# Properties

 Label 3800.1.bn.c Level $3800$ Weight $1$ Character orbit 3800.bn Analytic conductor $1.896$ Analytic rank $0$ Dimension $4$ Projective image $D_{3}$ CM discriminant -8 Inner twists $8$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$3800 = 2^{3} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3800.bn (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.89644704801$$ 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: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{3}$$ Projective field: Galois closure of 3.1.72200.2 Artin image: $C_{12}\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{24} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \zeta_{12}^{5} q^{2} - \zeta_{12}^{5} q^{3} - \zeta_{12}^{4} q^{4} - \zeta_{12}^{4} q^{6} - \zeta_{12}^{3} q^{8} +O(q^{10})$$ q - z^5 * q^2 - z^5 * q^3 - z^4 * q^4 - z^4 * q^6 - z^3 * q^8 $$q - \zeta_{12}^{5} q^{2} - \zeta_{12}^{5} q^{3} - \zeta_{12}^{4} q^{4} - \zeta_{12}^{4} q^{6} - \zeta_{12}^{3} q^{8} + q^{11} - \zeta_{12}^{3} q^{12} - \zeta_{12}^{2} q^{16} + \zeta_{12}^{5} q^{17} + \zeta_{12}^{2} q^{19} - 2 \zeta_{12}^{5} q^{22} - \zeta_{12}^{2} q^{24} + \zeta_{12}^{3} q^{27} - \zeta_{12} q^{32} - 2 \zeta_{12}^{5} q^{33} + \zeta_{12}^{4} q^{34} + \zeta_{12} q^{38} - \zeta_{12}^{2} q^{41} + \zeta_{12}^{5} q^{43} - 2 \zeta_{12}^{4} q^{44} - \zeta_{12} q^{48} - q^{49} + \zeta_{12}^{4} q^{51} + \zeta_{12}^{2} q^{54} + \zeta_{12} q^{57} - \zeta_{12}^{2} q^{59} - q^{64} - 2 \zeta_{12}^{4} q^{66} - \zeta_{12} q^{67} + \zeta_{12}^{3} q^{68} - \zeta_{12}^{5} q^{73} + q^{76} + \zeta_{12}^{2} q^{81} - 2 \zeta_{12} q^{82} + \zeta_{12}^{3} q^{83} + 2 \zeta_{12}^{4} q^{86} - 2 \zeta_{12}^{3} q^{88} + \zeta_{12}^{4} q^{89} - q^{96} + \zeta_{12}^{5} q^{97} + \zeta_{12}^{5} q^{98} +O(q^{100})$$ q - z^5 * q^2 - z^5 * q^3 - z^4 * q^4 - z^4 * q^6 - z^3 * q^8 + q^11 - z^3 * q^12 - z^2 * q^16 + z^5 * q^17 + z^2 * q^19 - 2*z^5 * q^22 - z^2 * q^24 + z^3 * q^27 - z * q^32 - 2*z^5 * q^33 + z^4 * q^34 + z * q^38 - z^2 * q^41 + z^5 * q^43 - 2*z^4 * q^44 - z * q^48 - q^49 + z^4 * q^51 + z^2 * q^54 + z * q^57 - z^2 * q^59 - q^64 - 2*z^4 * q^66 - z * q^67 + z^3 * q^68 - z^5 * q^73 + q^76 + z^2 * q^81 - 2*z * q^82 + z^3 * q^83 + 2*z^4 * q^86 - 2*z^3 * q^88 + z^4 * q^89 - q^96 + z^5 * q^97 + z^5 * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} + 2 q^{6}+O(q^{10})$$ 4 * q + 2 * q^4 + 2 * q^6 $$4 q + 2 q^{4} + 2 q^{6} + 8 q^{11} - 2 q^{16} + 2 q^{19} - 2 q^{24} - 2 q^{34} - 4 q^{41} + 4 q^{44} - 4 q^{49} - 2 q^{51} + 2 q^{54} - 2 q^{59} - 4 q^{64} + 4 q^{66} + 4 q^{76} + 2 q^{81} - 4 q^{86} - 2 q^{89} - 4 q^{96}+O(q^{100})$$ 4 * q + 2 * q^4 + 2 * q^6 + 8 * q^11 - 2 * q^16 + 2 * q^19 - 2 * q^24 - 2 * q^34 - 4 * q^41 + 4 * q^44 - 4 * q^49 - 2 * q^51 + 2 * q^54 - 2 * q^59 - 4 * q^64 + 4 * q^66 + 4 * q^76 + 2 * q^81 - 4 * q^86 - 2 * q^89 - 4 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3800\mathbb{Z}\right)^\times$$.

 $$n$$ $$401$$ $$951$$ $$1901$$ $$1977$$ $$\chi(n)$$ $$\zeta_{12}^{4}$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1299.1
 −0.866025 − 0.500000i 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 0.500000 0.866025i 0 1.00000i 0 0
1299.2 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i 0 0.500000 0.866025i 0 1.00000i 0 0
2899.1 −0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 1.00000i 0 0
2899.2 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 0 1.00000i 0 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
8.d odd 2 1 CM by $$\Q(\sqrt{-2})$$
5.b even 2 1 inner
19.c even 3 1 inner
40.e odd 2 1 inner
95.i even 6 1 inner
152.k odd 6 1 inner
760.bm odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3800.1.bn.c 4
5.b even 2 1 inner 3800.1.bn.c 4
5.c odd 4 1 3800.1.bd.b 2
5.c odd 4 1 3800.1.bd.d yes 2
8.d odd 2 1 CM 3800.1.bn.c 4
19.c even 3 1 inner 3800.1.bn.c 4
40.e odd 2 1 inner 3800.1.bn.c 4
40.k even 4 1 3800.1.bd.b 2
40.k even 4 1 3800.1.bd.d yes 2
95.i even 6 1 inner 3800.1.bn.c 4
95.m odd 12 1 3800.1.bd.b 2
95.m odd 12 1 3800.1.bd.d yes 2
152.k odd 6 1 inner 3800.1.bn.c 4
760.bm odd 6 1 inner 3800.1.bn.c 4
760.bw even 12 1 3800.1.bd.b 2
760.bw even 12 1 3800.1.bd.d yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3800.1.bd.b 2 5.c odd 4 1
3800.1.bd.b 2 40.k even 4 1
3800.1.bd.b 2 95.m odd 12 1
3800.1.bd.b 2 760.bw even 12 1
3800.1.bd.d yes 2 5.c odd 4 1
3800.1.bd.d yes 2 40.k even 4 1
3800.1.bd.d yes 2 95.m odd 12 1
3800.1.bd.d yes 2 760.bw even 12 1
3800.1.bn.c 4 1.a even 1 1 trivial
3800.1.bn.c 4 5.b even 2 1 inner
3800.1.bn.c 4 8.d odd 2 1 CM
3800.1.bn.c 4 19.c even 3 1 inner
3800.1.bn.c 4 40.e odd 2 1 inner
3800.1.bn.c 4 95.i even 6 1 inner
3800.1.bn.c 4 152.k odd 6 1 inner
3800.1.bn.c 4 760.bm odd 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{1}^{\mathrm{new}}(3800, [\chi])$$:

 $$T_{3}^{4} - T_{3}^{2} + 1$$ T3^4 - T3^2 + 1 $$T_{11} - 2$$ T11 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - T^{2} + 1$$
$3$ $$T^{4} - T^{2} + 1$$
$5$ $$T^{4}$$
$7$ $$T^{4}$$
$11$ $$(T - 2)^{4}$$
$13$ $$T^{4}$$
$17$ $$T^{4} - T^{2} + 1$$
$19$ $$(T^{2} - T + 1)^{2}$$
$23$ $$T^{4}$$
$29$ $$T^{4}$$
$31$ $$T^{4}$$
$37$ $$T^{4}$$
$41$ $$(T^{2} + 2 T + 4)^{2}$$
$43$ $$T^{4} - 4T^{2} + 16$$
$47$ $$T^{4}$$
$53$ $$T^{4}$$
$59$ $$(T^{2} + T + 1)^{2}$$
$61$ $$T^{4}$$
$67$ $$T^{4} - 4T^{2} + 16$$
$71$ $$T^{4}$$
$73$ $$T^{4} - T^{2} + 1$$
$79$ $$T^{4}$$
$83$ $$(T^{2} + 1)^{2}$$
$89$ $$(T^{2} + T + 1)^{2}$$
$97$ $$T^{4} - T^{2} + 1$$
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