# Properties

 Label 1400.2.q.j Level $1400$ Weight $2$ Character orbit 1400.q Analytic conductor $11.179$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1400.q (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.1790562830$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.11337408.1 Defining polynomial: $$x^{6} + 18 x^{4} + 81 x^{2} + 12$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 280) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{3} + \beta_{4} - \beta_{5} ) q^{3} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{7} + ( -\beta_{1} - 3 \beta_{2} + \beta_{4} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{3} + \beta_{4} - \beta_{5} ) q^{3} + ( -1 - \beta_{1} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{7} + ( -\beta_{1} - 3 \beta_{2} + \beta_{4} ) q^{9} + ( -1 + 2 \beta_{1} + \beta_{2} + \beta_{4} ) q^{11} + ( -1 - \beta_{1} + 2 \beta_{3} ) q^{13} + ( 2 - 2 \beta_{2} ) q^{17} + ( -\beta_{1} - \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{19} + ( 2 - \beta_{1} - 4 \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{21} + ( -\beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} ) q^{23} + ( 6 - 3 \beta_{3} - 3 \beta_{4} ) q^{27} + ( 4 + \beta_{1} + 2 \beta_{4} ) q^{29} + ( -4 + 4 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{31} + ( -6 \beta_{2} + 4 \beta_{5} ) q^{33} + ( -\beta_{1} + 3 \beta_{2} + \beta_{4} - 2 \beta_{5} ) q^{37} + ( 6 - 4 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} - 2 \beta_{5} ) q^{39} + ( 3 - 2 \beta_{1} + 2 \beta_{3} - 2 \beta_{4} ) q^{41} + ( -4 - \beta_{3} - \beta_{4} ) q^{43} + ( -3 \beta_{1} - 5 \beta_{2} + 3 \beta_{4} - 2 \beta_{5} ) q^{47} + ( -1 - 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} ) q^{49} -2 \beta_{5} q^{51} + ( 3 + 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{53} + ( -6 + 2 \beta_{1} - 4 \beta_{3} ) q^{57} + ( -8 + 8 \beta_{2} ) q^{59} + ( -\beta_{1} + 2 \beta_{2} + \beta_{4} + 2 \beta_{5} ) q^{61} + ( -6 + \beta_{1} + 9 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{63} + ( 2 - 2 \beta_{2} - 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{67} + ( \beta_{1} - 2 \beta_{3} ) q^{69} + ( 4 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} ) q^{73} + ( 7 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{77} + ( -2 \beta_{1} + 6 \beta_{2} + 2 \beta_{4} - 4 \beta_{5} ) q^{79} + ( -9 + 9 \beta_{2} + 6 \beta_{3} + 6 \beta_{4} - 6 \beta_{5} ) q^{81} + ( 10 + \beta_{3} + \beta_{4} ) q^{83} + ( 6 - 6 \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{87} + ( \beta_{1} - \beta_{4} + 2 \beta_{5} ) q^{89} + ( 9 + \beta_{1} - 10 \beta_{2} - 2 \beta_{5} ) q^{91} + ( 2 \beta_{1} + 12 \beta_{2} - 2 \beta_{4} + 4 \beta_{5} ) q^{93} + 2 q^{97} + ( 21 - \beta_{1} - 6 \beta_{3} - 8 \beta_{4} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6 q - 6 q^{7} - 9 q^{9} + O(q^{10})$$ $$6 q - 6 q^{7} - 9 q^{9} - 3 q^{11} - 6 q^{13} + 6 q^{17} - 3 q^{19} + 3 q^{23} + 36 q^{27} + 24 q^{29} - 12 q^{31} - 18 q^{33} + 9 q^{37} + 18 q^{39} + 18 q^{41} - 24 q^{43} - 15 q^{47} - 12 q^{49} + 9 q^{53} - 36 q^{57} - 24 q^{59} + 6 q^{61} - 9 q^{63} + 6 q^{67} + 39 q^{77} + 18 q^{79} - 27 q^{81} + 60 q^{83} + 18 q^{87} + 24 q^{91} + 36 q^{93} + 12 q^{97} + 126 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 18 x^{4} + 81 x^{2} + 12$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 9 \nu + 2$$$$)/4$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{2} + \nu + 6$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{4} + 9 \nu^{2} - 2 \nu$$$$)/4$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{5} + \nu^{4} + 15 \nu^{3} + 11 \nu^{2} + 48 \nu + 12$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{3} - \beta_{1} - 6$$ $$\nu^{3}$$ $$=$$ $$4 \beta_{2} - 9 \beta_{1} - 2$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{4} - 18 \beta_{3} + 11 \beta_{1} + 54$$ $$\nu^{5}$$ $$=$$ $$8 \beta_{5} - 4 \beta_{4} - 4 \beta_{3} - 60 \beta_{2} + 87 \beta_{1} + 30$$

## Character values

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

 $$n$$ $$351$$ $$701$$ $$801$$ $$1177$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
401.1
 − 2.78499i 3.17656i − 0.391571i 2.78499i − 3.17656i 0.391571i
0 −1.64497 2.84918i 0 0 0 −2.64497 0.0641892i 0 −3.91187 + 6.77556i 0
401.2 0 0.352860 + 0.611171i 0 0 0 −0.647140 2.56539i 0 1.25098 2.16676i 0
401.3 0 1.29211 + 2.23800i 0 0 0 0.292113 + 2.62958i 0 −1.83911 + 3.18543i 0
1201.1 0 −1.64497 + 2.84918i 0 0 0 −2.64497 + 0.0641892i 0 −3.91187 6.77556i 0
1201.2 0 0.352860 0.611171i 0 0 0 −0.647140 + 2.56539i 0 1.25098 + 2.16676i 0
1201.3 0 1.29211 2.23800i 0 0 0 0.292113 2.62958i 0 −1.83911 3.18543i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1201.3 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.q.j 6
5.b even 2 1 280.2.q.e 6
5.c odd 4 2 1400.2.bh.i 12
7.c even 3 1 inner 1400.2.q.j 6
7.c even 3 1 9800.2.a.ce 3
7.d odd 6 1 9800.2.a.cf 3
15.d odd 2 1 2520.2.bi.q 6
20.d odd 2 1 560.2.q.l 6
35.c odd 2 1 1960.2.q.w 6
35.i odd 6 1 1960.2.a.v 3
35.i odd 6 1 1960.2.q.w 6
35.j even 6 1 280.2.q.e 6
35.j even 6 1 1960.2.a.w 3
35.l odd 12 2 1400.2.bh.i 12
105.o odd 6 1 2520.2.bi.q 6
140.p odd 6 1 560.2.q.l 6
140.p odd 6 1 3920.2.a.cc 3
140.s even 6 1 3920.2.a.cb 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.q.e 6 5.b even 2 1
280.2.q.e 6 35.j even 6 1
560.2.q.l 6 20.d odd 2 1
560.2.q.l 6 140.p odd 6 1
1400.2.q.j 6 1.a even 1 1 trivial
1400.2.q.j 6 7.c even 3 1 inner
1400.2.bh.i 12 5.c odd 4 2
1400.2.bh.i 12 35.l odd 12 2
1960.2.a.v 3 35.i odd 6 1
1960.2.a.w 3 35.j even 6 1
1960.2.q.w 6 35.c odd 2 1
1960.2.q.w 6 35.i odd 6 1
2520.2.bi.q 6 15.d odd 2 1
2520.2.bi.q 6 105.o odd 6 1
3920.2.a.cb 3 140.s even 6 1
3920.2.a.cc 3 140.p odd 6 1
9800.2.a.ce 3 7.c even 3 1
9800.2.a.cf 3 7.d odd 6 1

## Hecke kernels

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

 $$T_{3}^{6} + 9 T_{3}^{4} - 12 T_{3}^{3} + 81 T_{3}^{2} - 54 T_{3} + 36$$ $$T_{11}^{6} + 3 T_{11}^{5} + 33 T_{11}^{4} + 16 T_{11}^{3} + 708 T_{11}^{2} + 1056 T_{11} + 1936$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$36 - 54 T + 81 T^{2} - 12 T^{3} + 9 T^{4} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$343 + 294 T + 168 T^{2} + 80 T^{3} + 24 T^{4} + 6 T^{5} + T^{6}$$
$11$ $$1936 + 1056 T + 708 T^{2} + 16 T^{3} + 33 T^{4} + 3 T^{5} + T^{6}$$
$13$ $$( -68 - 24 T + 3 T^{2} + T^{3} )^{2}$$
$17$ $$( 4 - 2 T + T^{2} )^{3}$$
$19$ $$256 - 384 T + 528 T^{2} - 104 T^{3} + 33 T^{4} + 3 T^{5} + T^{6}$$
$23$ $$49 + 105 T + 204 T^{2} + 59 T^{3} + 24 T^{4} - 3 T^{5} + T^{6}$$
$29$ $$( 26 + 21 T - 12 T^{2} + T^{3} )^{2}$$
$31$ $$16384 - 1536 T + 1680 T^{2} + 400 T^{3} + 132 T^{4} + 12 T^{5} + T^{6}$$
$37$ $$9216 + 864 T^{2} - 192 T^{3} + 81 T^{4} - 9 T^{5} + T^{6}$$
$41$ $$( 381 - 45 T - 9 T^{2} + T^{3} )^{2}$$
$43$ $$( 22 + 39 T + 12 T^{2} + T^{3} )^{2}$$
$47$ $$2521744 + 152448 T + 33036 T^{2} + 1736 T^{3} + 321 T^{4} + 15 T^{5} + T^{6}$$
$53$ $$389376 - 44928 T + 10800 T^{2} - 600 T^{3} + 153 T^{4} - 9 T^{5} + T^{6}$$
$59$ $$( 64 + 8 T + T^{2} )^{3}$$
$61$ $$295936 - 47328 T + 10833 T^{2} - 566 T^{3} + 123 T^{4} - 6 T^{5} + T^{6}$$
$67$ $$64 + 552 T + 4713 T^{2} + 430 T^{3} + 105 T^{4} - 6 T^{5} + T^{6}$$
$71$ $$T^{6}$$
$73$ $$112896 - 36288 T + 11664 T^{2} - 672 T^{3} + 108 T^{4} + T^{6}$$
$79$ $$589824 + 13824 T^{2} - 1536 T^{3} + 324 T^{4} - 18 T^{5} + T^{6}$$
$83$ $$( -904 + 291 T - 30 T^{2} + T^{3} )^{2}$$
$89$ $$1764 + 1134 T + 729 T^{2} + 84 T^{3} + 27 T^{4} + T^{6}$$
$97$ $$( -2 + T )^{6}$$