# Properties

 Label 3100.1.t.b Level $3100$ Weight $1$ Character orbit 3100.t Analytic conductor $1.547$ Analytic rank $0$ Dimension $4$ Projective image $A_{4}$ CM/RM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3100 = 2^{2} \cdot 5^{2} \cdot 31$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 3100.t (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$1.54710153916$$ 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: $$\Z[a_1, \ldots, a_{11}]$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{2} - \zeta_{12}^{2} q^{3} + q^{4} - \zeta_{12}^{2} q^{6} - \zeta_{12}^{2} q^{7} + q^{8} +O(q^{10})$$ q + q^2 - z^2 * q^3 + q^4 - z^2 * q^6 - z^2 * q^7 + q^8 $$q + q^{2} - \zeta_{12}^{2} q^{3} + q^{4} - \zeta_{12}^{2} q^{6} - \zeta_{12}^{2} q^{7} + q^{8} - \zeta_{12} q^{11} - \zeta_{12}^{2} q^{12} - \zeta_{12} q^{13} - \zeta_{12}^{2} q^{14} + q^{16} - \zeta_{12}^{5} q^{17} + \zeta_{12}^{5} q^{19} + \zeta_{12}^{4} q^{21} - \zeta_{12} q^{22} - \zeta_{12}^{2} q^{24} - \zeta_{12} q^{26} - q^{27} - \zeta_{12}^{2} q^{28} + \zeta_{12}^{3} q^{31} + q^{32} + \zeta_{12}^{3} q^{33} - \zeta_{12}^{5} q^{34} - \zeta_{12}^{5} q^{37} + \zeta_{12}^{5} q^{38} + \zeta_{12}^{3} q^{39} + \zeta_{12}^{4} q^{41} + \zeta_{12}^{4} q^{42} - \zeta_{12}^{2} q^{43} - \zeta_{12} q^{44} - \zeta_{12}^{2} q^{48} - \zeta_{12} q^{51} - \zeta_{12} q^{52} - \zeta_{12} q^{53} - q^{54} - \zeta_{12}^{2} q^{56} + \zeta_{12} q^{57} - \zeta_{12}^{5} q^{59} + \zeta_{12}^{3} q^{62} + q^{64} + \zeta_{12}^{3} q^{66} - \zeta_{12}^{4} q^{67} - \zeta_{12}^{5} q^{68} + \zeta_{12} q^{71} + \zeta_{12} q^{73} - \zeta_{12}^{5} q^{74} + \zeta_{12}^{5} q^{76} + \zeta_{12}^{3} q^{77} + \zeta_{12}^{3} q^{78} - \zeta_{12}^{5} q^{79} + \zeta_{12}^{2} q^{81} + \zeta_{12}^{4} q^{82} - \zeta_{12}^{4} q^{83} + \zeta_{12}^{4} q^{84} - \zeta_{12}^{2} q^{86} - \zeta_{12} q^{88} + \zeta_{12}^{3} q^{91} - \zeta_{12}^{5} q^{93} - \zeta_{12}^{2} q^{96} +O(q^{100})$$ q + q^2 - z^2 * q^3 + q^4 - z^2 * q^6 - z^2 * q^7 + q^8 - z * q^11 - z^2 * q^12 - z * q^13 - z^2 * q^14 + q^16 - z^5 * q^17 + z^5 * q^19 + z^4 * q^21 - z * q^22 - z^2 * q^24 - z * q^26 - q^27 - z^2 * q^28 + z^3 * q^31 + q^32 + z^3 * q^33 - z^5 * q^34 - z^5 * q^37 + z^5 * q^38 + z^3 * q^39 + z^4 * q^41 + z^4 * q^42 - z^2 * q^43 - z * q^44 - z^2 * q^48 - z * q^51 - z * q^52 - z * q^53 - q^54 - z^2 * q^56 + z * q^57 - z^5 * q^59 + z^3 * q^62 + q^64 + z^3 * q^66 - z^4 * q^67 - z^5 * q^68 + z * q^71 + z * q^73 - z^5 * q^74 + z^5 * q^76 + z^3 * q^77 + z^3 * q^78 - z^5 * q^79 + z^2 * q^81 + z^4 * q^82 - z^4 * q^83 + z^4 * q^84 - z^2 * q^86 - z * q^88 + z^3 * q^91 - z^5 * q^93 - z^2 * q^96 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{6} - 2 q^{7} + 4 q^{8}+O(q^{10})$$ 4 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^6 - 2 * q^7 + 4 * q^8 $$4 q + 4 q^{2} - 2 q^{3} + 4 q^{4} - 2 q^{6} - 2 q^{7} + 4 q^{8} - 2 q^{12} - 2 q^{14} + 4 q^{16} - 2 q^{21} - 2 q^{24} - 4 q^{27} - 2 q^{28} + 4 q^{32} - 2 q^{41} - 2 q^{42} - 2 q^{43} - 2 q^{48} - 4 q^{54} - 2 q^{56} + 4 q^{64} + 2 q^{67} + 2 q^{81} - 2 q^{82} + 2 q^{83} - 2 q^{84} - 2 q^{86} - 2 q^{96}+O(q^{100})$$ 4 * q + 4 * q^2 - 2 * q^3 + 4 * q^4 - 2 * q^6 - 2 * q^7 + 4 * q^8 - 2 * q^12 - 2 * q^14 + 4 * q^16 - 2 * q^21 - 2 * q^24 - 4 * q^27 - 2 * q^28 + 4 * q^32 - 2 * q^41 - 2 * q^42 - 2 * q^43 - 2 * q^48 - 4 * q^54 - 2 * q^56 + 4 * q^64 + 2 * q^67 + 2 * q^81 - 2 * q^82 + 2 * q^83 - 2 * q^84 - 2 * q^86 - 2 * q^96

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3100\mathbb{Z}\right)^\times$$.

 $$n$$ $$1551$$ $$1801$$ $$2977$$ $$\chi(n)$$ $$-1$$ $$\zeta_{12}^{4}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1699.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
1.00000 −0.500000 0.866025i 1.00000 0 −0.500000 0.866025i −0.500000 0.866025i 1.00000 0 0
1699.2 1.00000 −0.500000 0.866025i 1.00000 0 −0.500000 0.866025i −0.500000 0.866025i 1.00000 0 0
2299.1 1.00000 −0.500000 + 0.866025i 1.00000 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 0 0
2299.2 1.00000 −0.500000 + 0.866025i 1.00000 0 −0.500000 + 0.866025i −0.500000 + 0.866025i 1.00000 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
20.d odd 2 1 inner
31.c even 3 1 inner
620.o odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3100.1.t.b 4
4.b odd 2 1 3100.1.t.a 4
5.b even 2 1 3100.1.t.a 4
5.c odd 4 1 124.1.i.a 4
5.c odd 4 1 3100.1.z.a 4
15.e even 4 1 1116.1.x.a 4
20.d odd 2 1 inner 3100.1.t.b 4
20.e even 4 1 124.1.i.a 4
20.e even 4 1 3100.1.z.a 4
31.c even 3 1 inner 3100.1.t.b 4
40.i odd 4 1 1984.1.s.a 4
40.k even 4 1 1984.1.s.a 4
60.l odd 4 1 1116.1.x.a 4
124.i odd 6 1 3100.1.t.a 4
155.f even 4 1 3844.1.i.d 4
155.j even 6 1 3100.1.t.a 4
155.o odd 12 1 124.1.i.a 4
155.o odd 12 1 3100.1.z.a 4
155.o odd 12 1 3844.1.b.d 2
155.p even 12 1 3844.1.b.c 2
155.p even 12 1 3844.1.i.d 4
155.r even 20 4 3844.1.n.f 16
155.s odd 20 4 3844.1.n.e 16
155.w odd 60 4 3844.1.l.d 8
155.w odd 60 4 3844.1.n.e 16
155.x even 60 4 3844.1.l.c 8
155.x even 60 4 3844.1.n.f 16
465.be even 12 1 1116.1.x.a 4
620.m odd 4 1 3844.1.i.d 4
620.o odd 6 1 inner 3100.1.t.b 4
620.bc odd 12 1 3844.1.b.c 2
620.bc odd 12 1 3844.1.i.d 4
620.be even 12 1 124.1.i.a 4
620.be even 12 1 3100.1.z.a 4
620.be even 12 1 3844.1.b.d 2
620.bh odd 20 4 3844.1.n.f 16
620.bj even 20 4 3844.1.n.e 16
620.bt even 60 4 3844.1.l.d 8
620.bt even 60 4 3844.1.n.e 16
620.bv odd 60 4 3844.1.l.c 8
620.bv odd 60 4 3844.1.n.f 16
1240.ch even 12 1 1984.1.s.a 4
1240.cj odd 12 1 1984.1.s.a 4
1860.cf odd 12 1 1116.1.x.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
124.1.i.a 4 5.c odd 4 1
124.1.i.a 4 20.e even 4 1
124.1.i.a 4 155.o odd 12 1
124.1.i.a 4 620.be even 12 1
1116.1.x.a 4 15.e even 4 1
1116.1.x.a 4 60.l odd 4 1
1116.1.x.a 4 465.be even 12 1
1116.1.x.a 4 1860.cf odd 12 1
1984.1.s.a 4 40.i odd 4 1
1984.1.s.a 4 40.k even 4 1
1984.1.s.a 4 1240.ch even 12 1
1984.1.s.a 4 1240.cj odd 12 1
3100.1.t.a 4 4.b odd 2 1
3100.1.t.a 4 5.b even 2 1
3100.1.t.a 4 124.i odd 6 1
3100.1.t.a 4 155.j even 6 1
3100.1.t.b 4 1.a even 1 1 trivial
3100.1.t.b 4 20.d odd 2 1 inner
3100.1.t.b 4 31.c even 3 1 inner
3100.1.t.b 4 620.o odd 6 1 inner
3100.1.z.a 4 5.c odd 4 1
3100.1.z.a 4 20.e even 4 1
3100.1.z.a 4 155.o odd 12 1
3100.1.z.a 4 620.be even 12 1
3844.1.b.c 2 155.p even 12 1
3844.1.b.c 2 620.bc odd 12 1
3844.1.b.d 2 155.o odd 12 1
3844.1.b.d 2 620.be even 12 1
3844.1.i.d 4 155.f even 4 1
3844.1.i.d 4 155.p even 12 1
3844.1.i.d 4 620.m odd 4 1
3844.1.i.d 4 620.bc odd 12 1
3844.1.l.c 8 155.x even 60 4
3844.1.l.c 8 620.bv odd 60 4
3844.1.l.d 8 155.w odd 60 4
3844.1.l.d 8 620.bt even 60 4
3844.1.n.e 16 155.s odd 20 4
3844.1.n.e 16 155.w odd 60 4
3844.1.n.e 16 620.bj even 20 4
3844.1.n.e 16 620.bt even 60 4
3844.1.n.f 16 155.r even 20 4
3844.1.n.f 16 155.x even 60 4
3844.1.n.f 16 620.bh odd 20 4
3844.1.n.f 16 620.bv odd 60 4

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{2} + T_{3} + 1$$ acting on $$S_{1}^{\mathrm{new}}(3100, [\chi])$$.

## Hecke characteristic polynomials

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