Properties

 Label 1400.2.bh.d Level $1400$ Weight $2$ Character orbit 1400.bh Analytic conductor $11.179$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1400.bh (of order $$6$$, degree $$2$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$11.1790562830$$ 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: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 2 \zeta_{12} q^{3} + ( - \zeta_{12}^{3} + 3 \zeta_{12}) q^{7} + \zeta_{12}^{2} q^{9}+O(q^{10})$$ q + 2*z * q^3 + (-z^3 + 3*z) * q^7 + z^2 * q^9 $$q + 2 \zeta_{12} q^{3} + ( - \zeta_{12}^{3} + 3 \zeta_{12}) q^{7} + \zeta_{12}^{2} q^{9} + (4 \zeta_{12}^{2} - 4) q^{11} + 2 \zeta_{12}^{3} q^{13} + 3 \zeta_{12} q^{17} + (4 \zeta_{12}^{2} + 2) q^{21} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{23} - 4 \zeta_{12}^{3} q^{27} + 6 q^{29} + (9 \zeta_{12}^{2} - 9) q^{31} + (8 \zeta_{12}^{3} - 8 \zeta_{12}) q^{33} + (4 \zeta_{12}^{2} - 4) q^{39} + 5 q^{41} + 6 \zeta_{12}^{3} q^{43} + ( - 9 \zeta_{12}^{3} + 9 \zeta_{12}) q^{47} + (3 \zeta_{12}^{2} + 5) q^{49} + 6 \zeta_{12}^{2} q^{51} - 6 \zeta_{12} q^{53} + ( - 8 \zeta_{12}^{2} + 8) q^{59} - 8 \zeta_{12}^{2} q^{61} + (2 \zeta_{12}^{3} + \zeta_{12}) q^{63} - 14 \zeta_{12} q^{67} - 6 q^{69} + 11 q^{71} - 2 \zeta_{12} q^{73} + (12 \zeta_{12}^{3} - 8 \zeta_{12}) q^{77} + 9 \zeta_{12}^{2} q^{79} + ( - 11 \zeta_{12}^{2} + 11) q^{81} + 6 \zeta_{12}^{3} q^{83} + 12 \zeta_{12} q^{87} + 11 \zeta_{12}^{2} q^{89} + (6 \zeta_{12}^{2} - 4) q^{91} + (18 \zeta_{12}^{3} - 18 \zeta_{12}) q^{93} - 11 \zeta_{12}^{3} q^{97} - 4 q^{99} +O(q^{100})$$ q + 2*z * q^3 + (-z^3 + 3*z) * q^7 + z^2 * q^9 + (4*z^2 - 4) * q^11 + 2*z^3 * q^13 + 3*z * q^17 + (4*z^2 + 2) * q^21 + (3*z^3 - 3*z) * q^23 - 4*z^3 * q^27 + 6 * q^29 + (9*z^2 - 9) * q^31 + (8*z^3 - 8*z) * q^33 + (4*z^2 - 4) * q^39 + 5 * q^41 + 6*z^3 * q^43 + (-9*z^3 + 9*z) * q^47 + (3*z^2 + 5) * q^49 + 6*z^2 * q^51 - 6*z * q^53 + (-8*z^2 + 8) * q^59 - 8*z^2 * q^61 + (2*z^3 + z) * q^63 - 14*z * q^67 - 6 * q^69 + 11 * q^71 - 2*z * q^73 + (12*z^3 - 8*z) * q^77 + 9*z^2 * q^79 + (-11*z^2 + 11) * q^81 + 6*z^3 * q^83 + 12*z * q^87 + 11*z^2 * q^89 + (6*z^2 - 4) * q^91 + (18*z^3 - 18*z) * q^93 - 11*z^3 * q^97 - 4 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^9 $$4 q + 2 q^{9} - 8 q^{11} + 16 q^{21} + 24 q^{29} - 18 q^{31} - 8 q^{39} + 20 q^{41} + 26 q^{49} + 12 q^{51} + 16 q^{59} - 16 q^{61} - 24 q^{69} + 44 q^{71} + 18 q^{79} + 22 q^{81} + 22 q^{89} - 4 q^{91} - 16 q^{99}+O(q^{100})$$ 4 * q + 2 * q^9 - 8 * q^11 + 16 * q^21 + 24 * q^29 - 18 * q^31 - 8 * q^39 + 20 * q^41 + 26 * q^49 + 12 * q^51 + 16 * q^59 - 16 * q^61 - 24 * q^69 + 44 * q^71 + 18 * q^79 + 22 * q^81 + 22 * q^89 - 4 * q^91 - 16 * q^99

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times$$.

 $$n$$ $$351$$ $$701$$ $$801$$ $$1177$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{12}^{2}$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
249.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 −1.73205 1.00000i 0 0 0 −2.59808 0.500000i 0 0.500000 + 0.866025i 0
249.2 0 1.73205 + 1.00000i 0 0 0 2.59808 + 0.500000i 0 0.500000 + 0.866025i 0
849.1 0 −1.73205 + 1.00000i 0 0 0 −2.59808 + 0.500000i 0 0.500000 0.866025i 0
849.2 0 1.73205 1.00000i 0 0 0 2.59808 0.500000i 0 0.500000 0.866025i 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.bh.d 4
5.b even 2 1 inner 1400.2.bh.d 4
5.c odd 4 1 1400.2.q.b 2
5.c odd 4 1 1400.2.q.f yes 2
7.c even 3 1 inner 1400.2.bh.d 4
35.j even 6 1 inner 1400.2.bh.d 4
35.k even 12 1 9800.2.a.k 1
35.k even 12 1 9800.2.a.bl 1
35.l odd 12 1 1400.2.q.b 2
35.l odd 12 1 1400.2.q.f yes 2
35.l odd 12 1 9800.2.a.j 1
35.l odd 12 1 9800.2.a.bm 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.q.b 2 5.c odd 4 1
1400.2.q.b 2 35.l odd 12 1
1400.2.q.f yes 2 5.c odd 4 1
1400.2.q.f yes 2 35.l odd 12 1
1400.2.bh.d 4 1.a even 1 1 trivial
1400.2.bh.d 4 5.b even 2 1 inner
1400.2.bh.d 4 7.c even 3 1 inner
1400.2.bh.d 4 35.j even 6 1 inner
9800.2.a.j 1 35.l odd 12 1
9800.2.a.k 1 35.k even 12 1
9800.2.a.bl 1 35.k even 12 1
9800.2.a.bm 1 35.l odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1400, [\chi])$$:

 $$T_{3}^{4} - 4T_{3}^{2} + 16$$ T3^4 - 4*T3^2 + 16 $$T_{11}^{2} + 4T_{11} + 16$$ T11^2 + 4*T11 + 16

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 4T^{2} + 16$$
$5$ $$T^{4}$$
$7$ $$T^{4} - 13T^{2} + 49$$
$11$ $$(T^{2} + 4 T + 16)^{2}$$
$13$ $$(T^{2} + 4)^{2}$$
$17$ $$T^{4} - 9T^{2} + 81$$
$19$ $$T^{4}$$
$23$ $$T^{4} - 9T^{2} + 81$$
$29$ $$(T - 6)^{4}$$
$31$ $$(T^{2} + 9 T + 81)^{2}$$
$37$ $$T^{4}$$
$41$ $$(T - 5)^{4}$$
$43$ $$(T^{2} + 36)^{2}$$
$47$ $$T^{4} - 81T^{2} + 6561$$
$53$ $$T^{4} - 36T^{2} + 1296$$
$59$ $$(T^{2} - 8 T + 64)^{2}$$
$61$ $$(T^{2} + 8 T + 64)^{2}$$
$67$ $$T^{4} - 196 T^{2} + 38416$$
$71$ $$(T - 11)^{4}$$
$73$ $$T^{4} - 4T^{2} + 16$$
$79$ $$(T^{2} - 9 T + 81)^{2}$$
$83$ $$(T^{2} + 36)^{2}$$
$89$ $$(T^{2} - 11 T + 121)^{2}$$
$97$ $$(T^{2} + 121)^{2}$$