# Properties

 Label 2368.2.g.h Level $2368$ Weight $2$ Character orbit 2368.g Analytic conductor $18.909$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$2368 = 2^{6} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2368.g (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.9085751986$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{21})$$ Defining polynomial: $$x^{4} + 11x^{2} + 25$$ x^4 + 11*x^2 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3$$ Twist minimal: no (minimal twist has level 74) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - 1) q^{3} - \beta_1 q^{5} + 2 q^{7} + ( - \beta_{3} + 3) q^{9}+O(q^{10})$$ q + (b3 - 1) * q^3 - b1 * q^5 + 2 * q^7 + (-b3 + 3) * q^9 $$q + (\beta_{3} - 1) q^{3} - \beta_1 q^{5} + 2 q^{7} + ( - \beta_{3} + 3) q^{9} + ( - \beta_{3} + 2) q^{11} - \beta_{2} q^{13} + (\beta_{2} + 3 \beta_1) q^{15} - 2 \beta_{2} q^{17} + 2 \beta_1 q^{19} + (2 \beta_{3} - 2) q^{21} - \beta_{2} q^{23} + (3 \beta_{3} - 4) q^{25} - 5 q^{27} + \beta_{2} q^{29} + ( - 2 \beta_{2} + \beta_1) q^{31} + (2 \beta_{3} - 7) q^{33} - 2 \beta_1 q^{35} + ( - \beta_{2} + \beta_1 - 4) q^{37} + ( - 2 \beta_{2} - \beta_1) q^{39} + (\beta_{3} + 7) q^{41} + (2 \beta_{2} + 2 \beta_1) q^{43} + ( - \beta_{2} - 5 \beta_1) q^{45} + (2 \beta_{3} + 2) q^{47} - 3 q^{49} + ( - 4 \beta_{2} - 2 \beta_1) q^{51} + (2 \beta_{3} - 4) q^{53} + ( - \beta_{2} - 4 \beta_1) q^{55} + ( - 2 \beta_{2} - 6 \beta_1) q^{57} + 2 \beta_1 q^{59} + (4 \beta_{2} + 3 \beta_1) q^{61} + ( - 2 \beta_{3} + 6) q^{63} + 3 q^{65} + (3 \beta_{3} - 2) q^{67} + ( - 2 \beta_{2} - \beta_1) q^{69} + (4 \beta_{3} - 2) q^{71} + ( - 3 \beta_{3} + 4) q^{73} + ( - 4 \beta_{3} + 19) q^{75} + ( - 2 \beta_{3} + 4) q^{77} + (3 \beta_{2} + 4 \beta_1) q^{79} + ( - 2 \beta_{3} - 4) q^{81} + (4 \beta_{3} + 4) q^{83} + 6 q^{85} + (2 \beta_{2} + \beta_1) q^{87} + ( - 2 \beta_{2} - 2 \beta_1) q^{89} - 2 \beta_{2} q^{91} + ( - 5 \beta_{2} - 5 \beta_1) q^{93} + ( - 6 \beta_{3} + 18) q^{95} + (4 \beta_{2} + 2 \beta_1) q^{97} + ( - 4 \beta_{3} + 11) q^{99}+O(q^{100})$$ q + (b3 - 1) * q^3 - b1 * q^5 + 2 * q^7 + (-b3 + 3) * q^9 + (-b3 + 2) * q^11 - b2 * q^13 + (b2 + 3*b1) * q^15 - 2*b2 * q^17 + 2*b1 * q^19 + (2*b3 - 2) * q^21 - b2 * q^23 + (3*b3 - 4) * q^25 - 5 * q^27 + b2 * q^29 + (-2*b2 + b1) * q^31 + (2*b3 - 7) * q^33 - 2*b1 * q^35 + (-b2 + b1 - 4) * q^37 + (-2*b2 - b1) * q^39 + (b3 + 7) * q^41 + (2*b2 + 2*b1) * q^43 + (-b2 - 5*b1) * q^45 + (2*b3 + 2) * q^47 - 3 * q^49 + (-4*b2 - 2*b1) * q^51 + (2*b3 - 4) * q^53 + (-b2 - 4*b1) * q^55 + (-2*b2 - 6*b1) * q^57 + 2*b1 * q^59 + (4*b2 + 3*b1) * q^61 + (-2*b3 + 6) * q^63 + 3 * q^65 + (3*b3 - 2) * q^67 + (-2*b2 - b1) * q^69 + (4*b3 - 2) * q^71 + (-3*b3 + 4) * q^73 + (-4*b3 + 19) * q^75 + (-2*b3 + 4) * q^77 + (3*b2 + 4*b1) * q^79 + (-2*b3 - 4) * q^81 + (4*b3 + 4) * q^83 + 6 * q^85 + (2*b2 + b1) * q^87 + (-2*b2 - 2*b1) * q^89 - 2*b2 * q^91 + (-5*b2 - 5*b1) * q^93 + (-6*b3 + 18) * q^95 + (4*b2 + 2*b1) * q^97 + (-4*b3 + 11) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} + 8 q^{7} + 10 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 + 8 * q^7 + 10 * q^9 $$4 q - 2 q^{3} + 8 q^{7} + 10 q^{9} + 6 q^{11} - 4 q^{21} - 10 q^{25} - 20 q^{27} - 24 q^{33} - 16 q^{37} + 30 q^{41} + 12 q^{47} - 12 q^{49} - 12 q^{53} + 20 q^{63} + 12 q^{65} - 2 q^{67} + 10 q^{73} + 68 q^{75} + 12 q^{77} - 20 q^{81} + 24 q^{83} + 24 q^{85} + 60 q^{95} + 36 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 + 8 * q^7 + 10 * q^9 + 6 * q^11 - 4 * q^21 - 10 * q^25 - 20 * q^27 - 24 * q^33 - 16 * q^37 + 30 * q^41 + 12 * q^47 - 12 * q^49 - 12 * q^53 + 20 * q^63 + 12 * q^65 - 2 * q^67 + 10 * q^73 + 68 * q^75 + 12 * q^77 - 20 * q^81 + 24 * q^83 + 24 * q^85 + 60 * q^95 + 36 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 11x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$( \nu^{3} + \nu ) / 5$$ (v^3 + v) / 5 $$\beta_{2}$$ $$=$$ $$( 2\nu^{3} + 17\nu ) / 5$$ (2*v^3 + 17*v) / 5 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 6$$ v^2 + 6
 $$\nu$$ $$=$$ $$( \beta_{2} - 2\beta_1 ) / 3$$ (b2 - 2*b1) / 3 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 6$$ b3 - 6 $$\nu^{3}$$ $$=$$ $$( -\beta_{2} + 17\beta_1 ) / 3$$ (-b2 + 17*b1) / 3

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/2368\mathbb{Z}\right)^\times$$.

 $$n$$ $$705$$ $$1407$$ $$1925$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
961.1
 − 2.79129i 2.79129i − 1.79129i 1.79129i
0 −2.79129 0 3.79129i 0 2.00000 0 4.79129 0
961.2 0 −2.79129 0 3.79129i 0 2.00000 0 4.79129 0
961.3 0 1.79129 0 0.791288i 0 2.00000 0 0.208712 0
961.4 0 1.79129 0 0.791288i 0 2.00000 0 0.208712 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
37.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2368.2.g.h 4
4.b odd 2 1 2368.2.g.j 4
8.b even 2 1 592.2.g.c 4
8.d odd 2 1 74.2.b.a 4
24.f even 2 1 666.2.c.b 4
24.h odd 2 1 5328.2.h.m 4
37.b even 2 1 inner 2368.2.g.h 4
40.e odd 2 1 1850.2.d.e 4
40.k even 4 1 1850.2.c.g 4
40.k even 4 1 1850.2.c.h 4
148.b odd 2 1 2368.2.g.j 4
296.e even 2 1 592.2.g.c 4
296.h odd 2 1 74.2.b.a 4
296.j even 4 1 2738.2.a.h 2
296.j even 4 1 2738.2.a.k 2
888.c even 2 1 666.2.c.b 4
888.i odd 2 1 5328.2.h.m 4
1480.h odd 2 1 1850.2.d.e 4
1480.bh even 4 1 1850.2.c.g 4
1480.bh even 4 1 1850.2.c.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.b.a 4 8.d odd 2 1
74.2.b.a 4 296.h odd 2 1
592.2.g.c 4 8.b even 2 1
592.2.g.c 4 296.e even 2 1
666.2.c.b 4 24.f even 2 1
666.2.c.b 4 888.c even 2 1
1850.2.c.g 4 40.k even 4 1
1850.2.c.g 4 1480.bh even 4 1
1850.2.c.h 4 40.k even 4 1
1850.2.c.h 4 1480.bh even 4 1
1850.2.d.e 4 40.e odd 2 1
1850.2.d.e 4 1480.h odd 2 1
2368.2.g.h 4 1.a even 1 1 trivial
2368.2.g.h 4 37.b even 2 1 inner
2368.2.g.j 4 4.b odd 2 1
2368.2.g.j 4 148.b odd 2 1
2738.2.a.h 2 296.j even 4 1
2738.2.a.k 2 296.j even 4 1
5328.2.h.m 4 24.h odd 2 1
5328.2.h.m 4 888.i odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(2368, [\chi])$$:

 $$T_{3}^{2} + T_{3} - 5$$ T3^2 + T3 - 5 $$T_{5}^{4} + 15T_{5}^{2} + 9$$ T5^4 + 15*T5^2 + 9

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + T - 5)^{2}$$
$5$ $$T^{4} + 15T^{2} + 9$$
$7$ $$(T - 2)^{4}$$
$11$ $$(T^{2} - 3 T - 3)^{2}$$
$13$ $$T^{4} + 15T^{2} + 9$$
$17$ $$T^{4} + 60T^{2} + 144$$
$19$ $$T^{4} + 60T^{2} + 144$$
$23$ $$T^{4} + 15T^{2} + 9$$
$29$ $$T^{4} + 15T^{2} + 9$$
$31$ $$T^{4} + 99T^{2} + 2025$$
$37$ $$(T^{2} + 8 T + 37)^{2}$$
$41$ $$(T^{2} - 15 T + 51)^{2}$$
$43$ $$(T^{2} + 36)^{2}$$
$47$ $$(T^{2} - 6 T - 12)^{2}$$
$53$ $$(T^{2} + 6 T - 12)^{2}$$
$59$ $$T^{4} + 60T^{2} + 144$$
$61$ $$T^{4} + 231 T^{2} + 11025$$
$67$ $$(T^{2} + T - 47)^{2}$$
$71$ $$(T^{2} - 84)^{2}$$
$73$ $$(T^{2} - 5 T - 41)^{2}$$
$79$ $$T^{4} + 231 T^{2} + 11025$$
$83$ $$(T^{2} - 12 T - 48)^{2}$$
$89$ $$(T^{2} + 36)^{2}$$
$97$ $$T^{4} + 204T^{2} + 3600$$
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