# Properties

 Label 3844.1.i.d Level $3844$ Weight $1$ Character orbit 3844.i Analytic conductor $1.918$ Analytic rank $0$ Dimension $4$ Projective image $A_{4}$ CM/RM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3844 = 2^{2} \cdot 31^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3844.i (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.91840590856$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 124) Projective image: $$A_{4}$$ Projective field: Galois closure of 4.0.15376.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{12}^{3} q^{2} - \zeta_{12} q^{3} - q^{4} - \zeta_{12}^{2} q^{5} - \zeta_{12}^{4} q^{6} - \zeta_{12} q^{7} - \zeta_{12}^{3} q^{8} +O(q^{10})$$ q + z^3 * q^2 - z * q^3 - q^4 - z^2 * q^5 - z^4 * q^6 - z * q^7 - z^3 * q^8 $$q + \zeta_{12}^{3} q^{2} - \zeta_{12} q^{3} - q^{4} - \zeta_{12}^{2} q^{5} - \zeta_{12}^{4} q^{6} - \zeta_{12} q^{7} - \zeta_{12}^{3} q^{8} - \zeta_{12}^{5} q^{10} - \zeta_{12}^{5} q^{11} + \zeta_{12} q^{12} - \zeta_{12}^{2} q^{13} - \zeta_{12}^{4} q^{14} + \zeta_{12}^{3} q^{15} + q^{16} - \zeta_{12}^{4} q^{17} + \zeta_{12} q^{19} + \zeta_{12}^{2} q^{20} + \zeta_{12}^{2} q^{21} + \zeta_{12}^{2} q^{22} + \zeta_{12}^{4} q^{24} - \zeta_{12}^{5} q^{26} + \zeta_{12}^{3} q^{27} + \zeta_{12} q^{28} - q^{30} + \zeta_{12}^{3} q^{32} - q^{33} + \zeta_{12} q^{34} + \zeta_{12}^{3} q^{35} - \zeta_{12}^{4} q^{37} + \zeta_{12}^{4} q^{38} + \zeta_{12}^{3} q^{39} + \zeta_{12}^{5} q^{40} - \zeta_{12}^{2} q^{41} + \zeta_{12}^{5} q^{42} - \zeta_{12} q^{43} + \zeta_{12}^{5} q^{44} - \zeta_{12} q^{48} + \zeta_{12}^{5} q^{51} + \zeta_{12}^{2} q^{52} - \zeta_{12}^{2} q^{53} - q^{54} - \zeta_{12} q^{55} + \zeta_{12}^{4} q^{56} - \zeta_{12}^{2} q^{57} - \zeta_{12} q^{59} - \zeta_{12}^{3} q^{60} - q^{64} + \zeta_{12}^{4} q^{65} - \zeta_{12}^{3} q^{66} + \zeta_{12}^{5} q^{67} + \zeta_{12}^{4} q^{68} - q^{70} - \zeta_{12}^{5} q^{71} + \zeta_{12}^{2} q^{73} + \zeta_{12} q^{74} - \zeta_{12} q^{76} - q^{77} - q^{78} + \zeta_{12} q^{79} - \zeta_{12}^{2} q^{80} - \zeta_{12}^{4} q^{81} - \zeta_{12}^{5} q^{82} + \zeta_{12}^{5} q^{83} - \zeta_{12}^{2} q^{84} - q^{85} - \zeta_{12}^{4} q^{86} - \zeta_{12}^{2} q^{88} + \zeta_{12}^{3} q^{91} - \zeta_{12}^{3} q^{95} - \zeta_{12}^{4} q^{96} +O(q^{100})$$ q + z^3 * q^2 - z * q^3 - q^4 - z^2 * q^5 - z^4 * q^6 - z * q^7 - z^3 * q^8 - z^5 * q^10 - z^5 * q^11 + z * q^12 - z^2 * q^13 - z^4 * q^14 + z^3 * q^15 + q^16 - z^4 * q^17 + z * q^19 + z^2 * q^20 + z^2 * q^21 + z^2 * q^22 + z^4 * q^24 - z^5 * q^26 + z^3 * q^27 + z * q^28 - q^30 + z^3 * q^32 - q^33 + z * q^34 + z^3 * q^35 - z^4 * q^37 + z^4 * q^38 + z^3 * q^39 + z^5 * q^40 - z^2 * q^41 + z^5 * q^42 - z * q^43 + z^5 * q^44 - z * q^48 + z^5 * q^51 + z^2 * q^52 - z^2 * q^53 - q^54 - z * q^55 + z^4 * q^56 - z^2 * q^57 - z * q^59 - z^3 * q^60 - q^64 + z^4 * q^65 - z^3 * q^66 + z^5 * q^67 + z^4 * q^68 - q^70 - z^5 * q^71 + z^2 * q^73 + z * q^74 - z * q^76 - q^77 - q^78 + z * q^79 - z^2 * q^80 - z^4 * q^81 - z^5 * q^82 + z^5 * q^83 - z^2 * q^84 - q^85 - z^4 * q^86 - z^2 * q^88 + z^3 * q^91 - z^3 * q^95 - z^4 * q^96 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 2 q^{5} + 2 q^{6}+O(q^{10})$$ 4 * q - 4 * q^4 - 2 * q^5 + 2 * q^6 $$4 q - 4 q^{4} - 2 q^{5} + 2 q^{6} - 2 q^{13} + 2 q^{14} + 4 q^{16} + 2 q^{17} + 2 q^{20} + 2 q^{21} + 2 q^{22} - 2 q^{24} - 4 q^{30} - 4 q^{33} + 2 q^{37} - 2 q^{38} - 2 q^{41} + 2 q^{52} - 2 q^{53} - 4 q^{54} - 2 q^{56} - 2 q^{57} - 4 q^{64} - 2 q^{65} - 2 q^{68} - 4 q^{70} + 2 q^{73} - 4 q^{77} - 4 q^{78} - 2 q^{80} + 2 q^{81} - 2 q^{84} - 4 q^{85} + 2 q^{86} - 2 q^{88} + 2 q^{96}+O(q^{100})$$ 4 * q - 4 * q^4 - 2 * q^5 + 2 * q^6 - 2 * q^13 + 2 * q^14 + 4 * q^16 + 2 * q^17 + 2 * q^20 + 2 * q^21 + 2 * q^22 - 2 * q^24 - 4 * q^30 - 4 * q^33 + 2 * q^37 - 2 * q^38 - 2 * q^41 + 2 * q^52 - 2 * q^53 - 4 * q^54 - 2 * q^56 - 2 * q^57 - 4 * q^64 - 2 * q^65 - 2 * q^68 - 4 * q^70 + 2 * q^73 - 4 * q^77 - 4 * q^78 - 2 * q^80 + 2 * q^81 - 2 * q^84 - 4 * q^85 + 2 * q^86 - 2 * q^88 + 2 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3844\mathbb{Z}\right)^\times$$.

 $$n$$ $$1923$$ $$1925$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{12}^{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
439.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
1.00000i 0.866025 + 0.500000i −1.00000 −0.500000 0.866025i 0.500000 0.866025i 0.866025 + 0.500000i 1.00000i 0 −0.866025 + 0.500000i
439.2 1.00000i −0.866025 0.500000i −1.00000 −0.500000 0.866025i 0.500000 0.866025i −0.866025 0.500000i 1.00000i 0 0.866025 0.500000i
2443.1 1.00000i −0.866025 + 0.500000i −1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i −0.866025 + 0.500000i 1.00000i 0 0.866025 + 0.500000i
2443.2 1.00000i 0.866025 0.500000i −1.00000 −0.500000 + 0.866025i 0.500000 + 0.866025i 0.866025 0.500000i 1.00000i 0 −0.866025 0.500000i
 $$n$$: 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
4.b odd 2 1 inner
31.c even 3 1 inner
124.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3844.1.i.d 4
4.b odd 2 1 inner 3844.1.i.d 4
31.b odd 2 1 124.1.i.a 4
31.c even 3 1 3844.1.b.c 2
31.c even 3 1 inner 3844.1.i.d 4
31.d even 5 4 3844.1.n.f 16
31.e odd 6 1 124.1.i.a 4
31.e odd 6 1 3844.1.b.d 2
31.f odd 10 4 3844.1.n.e 16
31.g even 15 4 3844.1.l.c 8
31.g even 15 4 3844.1.n.f 16
31.h odd 30 4 3844.1.l.d 8
31.h odd 30 4 3844.1.n.e 16
93.c even 2 1 1116.1.x.a 4
93.g even 6 1 1116.1.x.a 4
124.d even 2 1 124.1.i.a 4
124.g even 6 1 124.1.i.a 4
124.g even 6 1 3844.1.b.d 2
124.i odd 6 1 3844.1.b.c 2
124.i odd 6 1 inner 3844.1.i.d 4
124.j even 10 4 3844.1.n.e 16
124.l odd 10 4 3844.1.n.f 16
124.n odd 30 4 3844.1.l.c 8
124.n odd 30 4 3844.1.n.f 16
124.p even 30 4 3844.1.l.d 8
124.p even 30 4 3844.1.n.e 16
155.c odd 2 1 3100.1.z.a 4
155.f even 4 1 3100.1.t.a 4
155.f even 4 1 3100.1.t.b 4
155.i odd 6 1 3100.1.z.a 4
155.p even 12 1 3100.1.t.a 4
155.p even 12 1 3100.1.t.b 4
248.b even 2 1 1984.1.s.a 4
248.g odd 2 1 1984.1.s.a 4
248.l odd 6 1 1984.1.s.a 4
248.q even 6 1 1984.1.s.a 4
372.b odd 2 1 1116.1.x.a 4
372.q odd 6 1 1116.1.x.a 4
620.e even 2 1 3100.1.z.a 4
620.m odd 4 1 3100.1.t.a 4
620.m odd 4 1 3100.1.t.b 4
620.r even 6 1 3100.1.z.a 4
620.bc odd 12 1 3100.1.t.a 4
620.bc odd 12 1 3100.1.t.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.1.i.a 4 31.b odd 2 1
124.1.i.a 4 31.e odd 6 1
124.1.i.a 4 124.d even 2 1
124.1.i.a 4 124.g even 6 1
1116.1.x.a 4 93.c even 2 1
1116.1.x.a 4 93.g even 6 1
1116.1.x.a 4 372.b odd 2 1
1116.1.x.a 4 372.q odd 6 1
1984.1.s.a 4 248.b even 2 1
1984.1.s.a 4 248.g odd 2 1
1984.1.s.a 4 248.l odd 6 1
1984.1.s.a 4 248.q even 6 1
3100.1.t.a 4 155.f even 4 1
3100.1.t.a 4 155.p even 12 1
3100.1.t.a 4 620.m odd 4 1
3100.1.t.a 4 620.bc odd 12 1
3100.1.t.b 4 155.f even 4 1
3100.1.t.b 4 155.p even 12 1
3100.1.t.b 4 620.m odd 4 1
3100.1.t.b 4 620.bc odd 12 1
3100.1.z.a 4 155.c odd 2 1
3100.1.z.a 4 155.i odd 6 1
3100.1.z.a 4 620.e even 2 1
3100.1.z.a 4 620.r even 6 1
3844.1.b.c 2 31.c even 3 1
3844.1.b.c 2 124.i odd 6 1
3844.1.b.d 2 31.e odd 6 1
3844.1.b.d 2 124.g even 6 1
3844.1.i.d 4 1.a even 1 1 trivial
3844.1.i.d 4 4.b odd 2 1 inner
3844.1.i.d 4 31.c even 3 1 inner
3844.1.i.d 4 124.i odd 6 1 inner
3844.1.l.c 8 31.g even 15 4
3844.1.l.c 8 124.n odd 30 4
3844.1.l.d 8 31.h odd 30 4
3844.1.l.d 8 124.p even 30 4
3844.1.n.e 16 31.f odd 10 4
3844.1.n.e 16 31.h odd 30 4
3844.1.n.e 16 124.j even 10 4
3844.1.n.e 16 124.p even 30 4
3844.1.n.f 16 31.d even 5 4
3844.1.n.f 16 31.g even 15 4
3844.1.n.f 16 124.l odd 10 4
3844.1.n.f 16 124.n odd 30 4

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3844, [\chi])$$:

 $$T_{3}^{4} - T_{3}^{2} + 1$$ T3^4 - T3^2 + 1 $$T_{5}^{2} + T_{5} + 1$$ T5^2 + T5 + 1 $$T_{7}^{4} - T_{7}^{2} + 1$$ T7^4 - T7^2 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4} - T^{2} + 1$$
$5$ $$(T^{2} + T + 1)^{2}$$
$7$ $$T^{4} - T^{2} + 1$$
$11$ $$T^{4} - T^{2} + 1$$
$13$ $$(T^{2} + T + 1)^{2}$$
$17$ $$(T^{2} - T + 1)^{2}$$
$19$ $$T^{4} - T^{2} + 1$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$(T^{2} - T + 1)^{2}$$
$41$ $$(T^{2} + T + 1)^{2}$$
$43$ $$T^{4} - T^{2} + 1$$
$47$ $$T^{4}$$
$53$ $$(T^{2} + T + 1)^{2}$$
$59$ $$T^{4} - T^{2} + 1$$
$61$ $$T^{4}$$
$67$ $$T^{4} - T^{2} + 1$$
$71$ $$T^{4} - T^{2} + 1$$
$73$ $$(T^{2} - T + 1)^{2}$$
$79$ $$T^{4} - T^{2} + 1$$
$83$ $$T^{4} - T^{2} + 1$$
$89$ $$T^{4}$$
$97$ $$T^{4}$$