# Properties

 Label 3844.1.l.c Level $3844$ Weight $1$ Character orbit 3844.l Analytic conductor $1.918$ Analytic rank $0$ Dimension $8$ Projective image $A_{4}$ CM/RM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3844 = 2^{2} \cdot 31^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3844.l (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.91840590856$$ Analytic rank: $$0$$ Dimension: $$8$$ Relative dimension: $$2$$ over $$\Q(\zeta_{10})$$ Coefficient field: $$\Q(\zeta_{20})$$ Defining polynomial: $$x^{8} - x^{6} + x^{4} - x^{2} + 1$$ x^8 - x^6 + x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ 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.

 $$f(q)$$ $$=$$ $$q - \zeta_{20}^{7} q^{2} - \zeta_{20}^{3} q^{3} - \zeta_{20}^{4} q^{4} + q^{5} - q^{6} + \zeta_{20}^{9} q^{7} - \zeta_{20} q^{8} +O(q^{10})$$ q - z^7 * q^2 - z^3 * q^3 - z^4 * q^4 + q^5 - q^6 + z^9 * q^7 - z * q^8 $$q - \zeta_{20}^{7} q^{2} - \zeta_{20}^{3} q^{3} - \zeta_{20}^{4} q^{4} + q^{5} - q^{6} + \zeta_{20}^{9} q^{7} - \zeta_{20} q^{8} - \zeta_{20}^{7} q^{10} + \zeta_{20}^{9} q^{11} + \zeta_{20}^{7} q^{12} + \zeta_{20}^{8} q^{13} + \zeta_{20}^{6} q^{14} - \zeta_{20}^{3} q^{15} + \zeta_{20}^{8} q^{16} + \zeta_{20}^{6} q^{17} + \zeta_{20}^{7} q^{19} - \zeta_{20}^{4} q^{20} + \zeta_{20}^{2} q^{21} + \zeta_{20}^{6} q^{22} + \zeta_{20}^{4} q^{24} + \zeta_{20}^{5} q^{26} + \zeta_{20}^{9} q^{27} + \zeta_{20}^{3} q^{28} - q^{30} + \zeta_{20}^{5} q^{32} + \zeta_{20}^{2} q^{33} + \zeta_{20}^{3} q^{34} + \zeta_{20}^{9} q^{35} - q^{37} + \zeta_{20}^{4} q^{38} + \zeta_{20} q^{39} - \zeta_{20} q^{40} - \zeta_{20}^{2} q^{41} - \zeta_{20}^{9} q^{42} - \zeta_{20}^{7} q^{43} + \zeta_{20}^{3} q^{44} + \zeta_{20} q^{48} - \zeta_{20}^{9} q^{51} + \zeta_{20}^{2} q^{52} - \zeta_{20}^{6} q^{53} + \zeta_{20}^{6} q^{54} + \zeta_{20}^{9} q^{55} + q^{56} + q^{57} - \zeta_{20}^{3} q^{59} + \zeta_{20}^{7} q^{60} + \zeta_{20}^{2} q^{64} + \zeta_{20}^{8} q^{65} - \zeta_{20}^{9} q^{66} - \zeta_{20}^{5} q^{67} + q^{68} + \zeta_{20}^{6} q^{70} + \zeta_{20} q^{71} - \zeta_{20}^{4} q^{73} + \zeta_{20}^{7} q^{74} + \zeta_{20} q^{76} - \zeta_{20}^{8} q^{77} - \zeta_{20}^{8} q^{78} - \zeta_{20} q^{79} + \zeta_{20}^{8} q^{80} + \zeta_{20}^{2} q^{81} + \zeta_{20}^{9} q^{82} + \zeta_{20}^{7} q^{83} - \zeta_{20}^{6} q^{84} + \zeta_{20}^{6} q^{85} - \zeta_{20}^{4} q^{86} + q^{88} - \zeta_{20}^{7} q^{91} + \zeta_{20}^{7} q^{95} - \zeta_{20}^{8} q^{96} +O(q^{100})$$ q - z^7 * q^2 - z^3 * q^3 - z^4 * q^4 + q^5 - q^6 + z^9 * q^7 - z * q^8 - z^7 * q^10 + z^9 * q^11 + z^7 * q^12 + z^8 * q^13 + z^6 * q^14 - z^3 * q^15 + z^8 * q^16 + z^6 * q^17 + z^7 * q^19 - z^4 * q^20 + z^2 * q^21 + z^6 * q^22 + z^4 * q^24 + z^5 * q^26 + z^9 * q^27 + z^3 * q^28 - q^30 + z^5 * q^32 + z^2 * q^33 + z^3 * q^34 + z^9 * q^35 - q^37 + z^4 * q^38 + z * q^39 - z * q^40 - z^2 * q^41 - z^9 * q^42 - z^7 * q^43 + z^3 * q^44 + z * q^48 - z^9 * q^51 + z^2 * q^52 - z^6 * q^53 + z^6 * q^54 + z^9 * q^55 + q^56 + q^57 - z^3 * q^59 + z^7 * q^60 + z^2 * q^64 + z^8 * q^65 - z^9 * q^66 - z^5 * q^67 + q^68 + z^6 * q^70 + z * q^71 - z^4 * q^73 + z^7 * q^74 + z * q^76 - z^8 * q^77 - z^8 * q^78 - z * q^79 + z^8 * q^80 + z^2 * q^81 + z^9 * q^82 + z^7 * q^83 - z^6 * q^84 + z^6 * q^85 - z^4 * q^86 + q^88 - z^7 * q^91 + z^7 * q^95 - z^8 * q^96 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8 q + 2 q^{4} + 8 q^{5} - 8 q^{6}+O(q^{10})$$ 8 * q + 2 * q^4 + 8 * q^5 - 8 * q^6 $$8 q + 2 q^{4} + 8 q^{5} - 8 q^{6} - 2 q^{13} + 2 q^{14} - 2 q^{16} + 2 q^{17} + 2 q^{20} + 2 q^{21} + 2 q^{22} - 2 q^{24} - 8 q^{30} + 2 q^{33} - 8 q^{37} - 2 q^{38} - 2 q^{41} + 2 q^{52} - 2 q^{53} + 2 q^{54} + 8 q^{56} + 8 q^{57} + 2 q^{64} - 2 q^{65} + 8 q^{68} + 2 q^{70} + 2 q^{73} + 2 q^{77} + 2 q^{78} - 2 q^{80} + 2 q^{81} - 2 q^{84} + 2 q^{85} + 2 q^{86} + 8 q^{88} + 2 q^{96}+O(q^{100})$$ 8 * q + 2 * q^4 + 8 * q^5 - 8 * q^6 - 2 * q^13 + 2 * q^14 - 2 * q^16 + 2 * q^17 + 2 * q^20 + 2 * q^21 + 2 * q^22 - 2 * q^24 - 8 * q^30 + 2 * q^33 - 8 * q^37 - 2 * q^38 - 2 * q^41 + 2 * q^52 - 2 * q^53 + 2 * q^54 + 8 * q^56 + 8 * q^57 + 2 * q^64 - 2 * q^65 + 8 * q^68 + 2 * q^70 + 2 * q^73 + 2 * q^77 + 2 * q^78 - 2 * q^80 + 2 * q^81 - 2 * q^84 + 2 * q^85 + 2 * q^86 + 8 * q^88 + 2 * q^96

## Character values

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

 $$n$$ $$1923$$ $$1925$$ $$\chi(n)$$ $$-1$$ $$-\zeta_{20}^{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
531.1
 0.587785 − 0.809017i −0.587785 + 0.809017i −0.951057 − 0.309017i 0.951057 + 0.309017i −0.951057 + 0.309017i 0.951057 − 0.309017i 0.587785 + 0.809017i −0.587785 − 0.809017i
−0.951057 + 0.309017i 0.951057 + 0.309017i 0.809017 0.587785i 1.00000 −1.00000 −0.587785 0.809017i −0.587785 + 0.809017i 0 −0.951057 + 0.309017i
531.2 0.951057 0.309017i −0.951057 0.309017i 0.809017 0.587785i 1.00000 −1.00000 0.587785 + 0.809017i 0.587785 0.809017i 0 0.951057 0.309017i
1335.1 −0.587785 + 0.809017i 0.587785 + 0.809017i −0.309017 0.951057i 1.00000 −1.00000 0.951057 0.309017i 0.951057 + 0.309017i 0 −0.587785 + 0.809017i
1335.2 0.587785 0.809017i −0.587785 0.809017i −0.309017 0.951057i 1.00000 −1.00000 −0.951057 + 0.309017i −0.951057 0.309017i 0 0.587785 0.809017i
3271.1 −0.587785 0.809017i 0.587785 0.809017i −0.309017 + 0.951057i 1.00000 −1.00000 0.951057 + 0.309017i 0.951057 0.309017i 0 −0.587785 0.809017i
3271.2 0.587785 + 0.809017i −0.587785 + 0.809017i −0.309017 + 0.951057i 1.00000 −1.00000 −0.951057 0.309017i −0.951057 + 0.309017i 0 0.587785 + 0.809017i
3511.1 −0.951057 0.309017i 0.951057 0.309017i 0.809017 + 0.587785i 1.00000 −1.00000 −0.587785 + 0.809017i −0.587785 0.809017i 0 −0.951057 0.309017i
3511.2 0.951057 + 0.309017i −0.951057 + 0.309017i 0.809017 + 0.587785i 1.00000 −1.00000 0.587785 0.809017i 0.587785 + 0.809017i 0 0.951057 + 0.309017i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 3511.2 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.d even 5 3 inner
124.l odd 10 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3844.1.l.c 8
4.b odd 2 1 inner 3844.1.l.c 8
31.b odd 2 1 3844.1.l.d 8
31.c even 3 2 3844.1.n.f 16
31.d even 5 1 3844.1.b.c 2
31.d even 5 3 inner 3844.1.l.c 8
31.e odd 6 2 3844.1.n.e 16
31.f odd 10 1 3844.1.b.d 2
31.f odd 10 3 3844.1.l.d 8
31.g even 15 2 3844.1.i.d 4
31.g even 15 6 3844.1.n.f 16
31.h odd 30 2 124.1.i.a 4
31.h odd 30 6 3844.1.n.e 16
93.p even 30 2 1116.1.x.a 4
124.d even 2 1 3844.1.l.d 8
124.g even 6 2 3844.1.n.e 16
124.i odd 6 2 3844.1.n.f 16
124.j even 10 1 3844.1.b.d 2
124.j even 10 3 3844.1.l.d 8
124.l odd 10 1 3844.1.b.c 2
124.l odd 10 3 inner 3844.1.l.c 8
124.n odd 30 2 3844.1.i.d 4
124.n odd 30 6 3844.1.n.f 16
124.p even 30 2 124.1.i.a 4
124.p even 30 6 3844.1.n.e 16
155.v odd 30 2 3100.1.z.a 4
155.x even 60 2 3100.1.t.a 4
155.x even 60 2 3100.1.t.b 4
248.bb even 30 2 1984.1.s.a 4
248.bf odd 30 2 1984.1.s.a 4
372.bc odd 30 2 1116.1.x.a 4
620.bo even 30 2 3100.1.z.a 4
620.bv odd 60 2 3100.1.t.a 4
620.bv odd 60 2 3100.1.t.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.1.i.a 4 31.h odd 30 2
124.1.i.a 4 124.p even 30 2
1116.1.x.a 4 93.p even 30 2
1116.1.x.a 4 372.bc odd 30 2
1984.1.s.a 4 248.bb even 30 2
1984.1.s.a 4 248.bf odd 30 2
3100.1.t.a 4 155.x even 60 2
3100.1.t.a 4 620.bv odd 60 2
3100.1.t.b 4 155.x even 60 2
3100.1.t.b 4 620.bv odd 60 2
3100.1.z.a 4 155.v odd 30 2
3100.1.z.a 4 620.bo even 30 2
3844.1.b.c 2 31.d even 5 1
3844.1.b.c 2 124.l odd 10 1
3844.1.b.d 2 31.f odd 10 1
3844.1.b.d 2 124.j even 10 1
3844.1.i.d 4 31.g even 15 2
3844.1.i.d 4 124.n odd 30 2
3844.1.l.c 8 1.a even 1 1 trivial
3844.1.l.c 8 4.b odd 2 1 inner
3844.1.l.c 8 31.d even 5 3 inner
3844.1.l.c 8 124.l odd 10 3 inner
3844.1.l.d 8 31.b odd 2 1
3844.1.l.d 8 31.f odd 10 3
3844.1.l.d 8 124.d even 2 1
3844.1.l.d 8 124.j even 10 3
3844.1.n.e 16 31.e odd 6 2
3844.1.n.e 16 31.h odd 30 6
3844.1.n.e 16 124.g even 6 2
3844.1.n.e 16 124.p even 30 6
3844.1.n.f 16 31.c even 3 2
3844.1.n.f 16 31.g even 15 6
3844.1.n.f 16 124.i odd 6 2
3844.1.n.f 16 124.n odd 30 6

## Hecke kernels

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

 $$T_{3}^{8} - T_{3}^{6} + T_{3}^{4} - T_{3}^{2} + 1$$ T3^8 - T3^6 + T3^4 - T3^2 + 1 $$T_{5} - 1$$ T5 - 1 $$T_{13}^{4} + T_{13}^{3} + T_{13}^{2} + T_{13} + 1$$ T13^4 + T13^3 + T13^2 + T13 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{8} - T^{6} + T^{4} - T^{2} + 1$$
$3$ $$T^{8} - T^{6} + T^{4} - T^{2} + 1$$
$5$ $$(T - 1)^{8}$$
$7$ $$T^{8} - T^{6} + T^{4} - T^{2} + 1$$
$11$ $$T^{8} - T^{6} + T^{4} - T^{2} + 1$$
$13$ $$(T^{4} + T^{3} + T^{2} + T + 1)^{2}$$
$17$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{2}$$
$19$ $$T^{8} - T^{6} + T^{4} - T^{2} + 1$$
$23$ $$T^{8}$$
$29$ $$T^{8}$$
$31$ $$T^{8}$$
$37$ $$(T + 1)^{8}$$
$41$ $$(T^{4} + T^{3} + T^{2} + T + 1)^{2}$$
$43$ $$T^{8} - T^{6} + T^{4} - T^{2} + 1$$
$47$ $$T^{8}$$
$53$ $$(T^{4} + T^{3} + T^{2} + T + 1)^{2}$$
$59$ $$T^{8} - T^{6} + T^{4} - T^{2} + 1$$
$61$ $$T^{8}$$
$67$ $$(T^{2} + 1)^{4}$$
$71$ $$T^{8} - T^{6} + T^{4} - T^{2} + 1$$
$73$ $$(T^{4} - T^{3} + T^{2} - T + 1)^{2}$$
$79$ $$T^{8} - T^{6} + T^{4} - T^{2} + 1$$
$83$ $$T^{8} - T^{6} + T^{4} - T^{2} + 1$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$