# Properties

 Label 1400.2.x.a Level $1400$ Weight $2$ Character orbit 1400.x Analytic conductor $11.179$ Analytic rank $0$ Dimension $16$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1400 = 2^{3} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1400.x (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.1790562830$$ Analytic rank: $$0$$ Dimension: $$16$$ Relative dimension: $$8$$ over $$\Q(i)$$ Coefficient field: 16.0.29960650073923649536.7 Defining polynomial: $$x^{16} - 7 x^{12} + 40 x^{8} - 112 x^{4} + 256$$ Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{22}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{15}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{6} q^{3} -\beta_{1} q^{7} -\beta_{4} q^{9} +O(q^{10})$$ $$q + \beta_{6} q^{3} -\beta_{1} q^{7} -\beta_{4} q^{9} + ( -1 - \beta_{9} ) q^{11} + ( \beta_{5} + \beta_{6} - \beta_{14} ) q^{13} + ( -\beta_{1} - \beta_{3} + \beta_{12} ) q^{17} + ( \beta_{11} + \beta_{13} ) q^{19} + ( -1 + \beta_{7} - \beta_{9} ) q^{21} + ( -\beta_{5} - \beta_{10} - \beta_{14} ) q^{23} + ( \beta_{1} - \beta_{3} - \beta_{12} ) q^{27} + ( \beta_{2} - 3 \beta_{4} ) q^{29} + ( -\beta_{7} - \beta_{8} ) q^{31} + ( \beta_{5} + \beta_{6} - \beta_{14} ) q^{33} -\beta_{15} q^{37} + ( -\beta_{2} - 3 \beta_{4} ) q^{39} + ( \beta_{7} - \beta_{8} ) q^{41} + ( -\beta_{5} + \beta_{10} - \beta_{14} ) q^{43} -\beta_{3} q^{47} + ( -\beta_{2} - 2 \beta_{4} + \beta_{13} ) q^{49} + ( 1 - 3 \beta_{9} ) q^{51} + ( 2 \beta_{5} + 2 \beta_{14} ) q^{53} + ( 2 \beta_{1} + 2 \beta_{12} + \beta_{15} ) q^{57} + 2 \beta_{11} q^{59} + ( \beta_{7} + \beta_{8} ) q^{61} + ( -\beta_{5} + \beta_{6} + \beta_{10} ) q^{63} + ( -\beta_{1} - \beta_{12} ) q^{67} + ( \beta_{11} + \beta_{13} ) q^{69} + ( -4 - 4 \beta_{9} ) q^{71} -4 \beta_{6} q^{73} + ( \beta_{1} - \beta_{3} - \beta_{12} + \beta_{15} ) q^{77} + ( \beta_{2} - 5 \beta_{4} ) q^{79} + ( 5 - 2 \beta_{9} ) q^{81} + ( -3 \beta_{5} - 4 \beta_{6} + 3 \beta_{14} ) q^{83} + ( 3 \beta_{1} + 5 \beta_{3} - 3 \beta_{12} ) q^{87} + ( -\beta_{11} + \beta_{13} ) q^{89} + ( 5 + \beta_{7} + \beta_{8} + \beta_{9} ) q^{91} + ( 2 \beta_{5} - 3 \beta_{10} + 2 \beta_{14} ) q^{93} + ( -3 \beta_{1} + 5 \beta_{3} + 3 \beta_{12} ) q^{97} -4 \beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$16q + O(q^{10})$$ $$16q - 8q^{11} - 8q^{21} + 40q^{51} - 32q^{71} + 96q^{81} + 72q^{91} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{16} - 7 x^{12} + 40 x^{8} - 112 x^{4} + 256$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{13} - 5 \nu^{9} - 36 \nu^{5} + 128 \nu$$$$)/128$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{10} - 3 \nu^{6} + 12 \nu^{2}$$$$)/16$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{9} + 3 \nu^{5} - 20 \nu$$$$)/8$$ $$\beta_{4}$$ $$=$$ $$($$$$\nu^{14} - 3 \nu^{10} + 28 \nu^{6}$$$$)/64$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{15} - 13 \nu^{11} + 52 \nu^{7} - 416 \nu^{3}$$$$)/256$$ $$\beta_{6}$$ $$=$$ $$($$$$-\nu^{15} - \nu^{11} - 16 \nu^{7} + 16 \nu^{3}$$$$)/128$$ $$\beta_{7}$$ $$=$$ $$($$$$3 \nu^{12} - \nu^{8} + 28 \nu^{4} + 64$$$$)/32$$ $$\beta_{8}$$ $$=$$ $$($$$$\nu^{12} - 11 \nu^{8} + 84 \nu^{4} - 192$$$$)/32$$ $$\beta_{9}$$ $$=$$ $$($$$$\nu^{12} - 7 \nu^{8} + 24 \nu^{4} - 64$$$$)/16$$ $$\beta_{10}$$ $$=$$ $$($$$$-\nu^{15} + 3 \nu^{11} + 4 \nu^{7} + 32 \nu^{3}$$$$)/64$$ $$\beta_{11}$$ $$=$$ $$($$$$-3 \nu^{14} + 17 \nu^{10} - 76 \nu^{6} + 256 \nu^{2}$$$$)/64$$ $$\beta_{12}$$ $$=$$ $$($$$$-5 \nu^{13} + 7 \nu^{9} - 20 \nu^{5}$$$$)/128$$ $$\beta_{13}$$ $$=$$ $$($$$$5 \nu^{14} - 23 \nu^{10} + 68 \nu^{6} - 64 \nu^{2}$$$$)/64$$ $$\beta_{14}$$ $$=$$ $$($$$$7 \nu^{15} - 53 \nu^{11} + 196 \nu^{7} - 480 \nu^{3}$$$$)/256$$ $$\beta_{15}$$ $$=$$ $$($$$$3 \nu^{13} - 17 \nu^{9} + 76 \nu^{5} - 128 \nu$$$$)/32$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{15} + 2 \beta_{12} - 4 \beta_{3} + 2 \beta_{1}$$$$)/8$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{13} + 3 \beta_{11} + 4 \beta_{4} - 4 \beta_{2}$$$$)/8$$ $$\nu^{3}$$ $$=$$ $$($$$$2 \beta_{14} + 3 \beta_{10} + 4 \beta_{6} - 6 \beta_{5}$$$$)/8$$ $$\nu^{4}$$ $$=$$ $$($$$$-4 \beta_{9} + 5 \beta_{8} + \beta_{7} + 12$$$$)/8$$ $$\nu^{5}$$ $$=$$ $$($$$$5 \beta_{15} + 14 \beta_{12} - 12 \beta_{3} - 10 \beta_{1}$$$$)/8$$ $$\nu^{6}$$ $$=$$ $$($$$$-5 \beta_{13} + \beta_{11} + 28 \beta_{4} - 12 \beta_{2}$$$$)/8$$ $$\nu^{7}$$ $$=$$ $$($$$$6 \beta_{14} + 21 \beta_{10} - 20 \beta_{6} - 2 \beta_{5}$$$$)/8$$ $$\nu^{8}$$ $$=$$ $$($$$$-28 \beta_{9} + 11 \beta_{8} + 15 \beta_{7} - 76$$$$)/8$$ $$\nu^{9}$$ $$=$$ $$($$$$-5 \beta_{15} + 2 \beta_{12} - 20 \beta_{3} - 70 \beta_{1}$$$$)/8$$ $$\nu^{10}$$ $$=$$ $$($$$$-27 \beta_{13} - 33 \beta_{11} + 36 \beta_{4} + 140 \beta_{2}$$$$)/8$$ $$\nu^{11}$$ $$=$$ $$($$$$-38 \beta_{14} + 11 \beta_{10} - 172 \beta_{6} + 34 \beta_{5}$$$$)/8$$ $$\nu^{12}$$ $$=$$ $$($$$$28 \beta_{9} - 43 \beta_{8} + 81 \beta_{7} - 308$$$$)/8$$ $$\nu^{13}$$ $$=$$ $$($$$$-27 \beta_{15} - 258 \beta_{12} + 20 \beta_{3} - 58 \beta_{1}$$$$)/8$$ $$\nu^{14}$$ $$=$$ $$($$$$59 \beta_{13} - 127 \beta_{11} - 164 \beta_{4} + 756 \beta_{2}$$$$)/8$$ $$\nu^{15}$$ $$=$$ $$($$$$-26 \beta_{14} - 299 \beta_{10} - 468 \beta_{6} - 98 \beta_{5}$$$$)/8$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
657.1
 −0.281691 + 1.38588i −1.38588 + 0.281691i 0.481610 + 1.32968i −1.32968 − 0.481610i 1.32968 + 0.481610i −0.481610 − 1.32968i 1.38588 − 0.281691i 0.281691 − 1.38588i −0.281691 − 1.38588i −1.38588 − 0.281691i 0.481610 − 1.32968i −1.32968 + 0.481610i 1.32968 − 0.481610i −0.481610 + 1.32968i 1.38588 + 0.281691i 0.281691 + 1.38588i
0 −1.66757 1.66757i 0 0 0 −2.27950 1.34308i 0 2.56155i 0
657.2 0 −1.66757 1.66757i 0 0 0 1.34308 + 2.27950i 0 2.56155i 0
657.3 0 −0.848071 0.848071i 0 0 0 0.406039 2.61441i 0 1.56155i 0
657.4 0 −0.848071 0.848071i 0 0 0 2.61441 0.406039i 0 1.56155i 0
657.5 0 0.848071 + 0.848071i 0 0 0 −2.61441 + 0.406039i 0 1.56155i 0
657.6 0 0.848071 + 0.848071i 0 0 0 −0.406039 + 2.61441i 0 1.56155i 0
657.7 0 1.66757 + 1.66757i 0 0 0 −1.34308 2.27950i 0 2.56155i 0
657.8 0 1.66757 + 1.66757i 0 0 0 2.27950 + 1.34308i 0 2.56155i 0
993.1 0 −1.66757 + 1.66757i 0 0 0 −2.27950 + 1.34308i 0 2.56155i 0
993.2 0 −1.66757 + 1.66757i 0 0 0 1.34308 2.27950i 0 2.56155i 0
993.3 0 −0.848071 + 0.848071i 0 0 0 0.406039 + 2.61441i 0 1.56155i 0
993.4 0 −0.848071 + 0.848071i 0 0 0 2.61441 + 0.406039i 0 1.56155i 0
993.5 0 0.848071 0.848071i 0 0 0 −2.61441 0.406039i 0 1.56155i 0
993.6 0 0.848071 0.848071i 0 0 0 −0.406039 2.61441i 0 1.56155i 0
993.7 0 1.66757 1.66757i 0 0 0 −1.34308 + 2.27950i 0 2.56155i 0
993.8 0 1.66757 1.66757i 0 0 0 2.27950 1.34308i 0 2.56155i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 993.8 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
5.c odd 4 2 inner
7.b odd 2 1 inner
35.c odd 2 1 inner
35.f even 4 2 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1400.2.x.a 16
5.b even 2 1 inner 1400.2.x.a 16
5.c odd 4 2 inner 1400.2.x.a 16
7.b odd 2 1 inner 1400.2.x.a 16
35.c odd 2 1 inner 1400.2.x.a 16
35.f even 4 2 inner 1400.2.x.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1400.2.x.a 16 1.a even 1 1 trivial
1400.2.x.a 16 5.b even 2 1 inner
1400.2.x.a 16 5.c odd 4 2 inner
1400.2.x.a 16 7.b odd 2 1 inner
1400.2.x.a 16 35.c odd 2 1 inner
1400.2.x.a 16 35.f even 4 2 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{8} + 33 T_{3}^{4} + 64$$ acting on $$S_{2}^{\mathrm{new}}(1400, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{16}$$
$3$ $$( 64 + 33 T^{4} + T^{8} )^{2}$$
$5$ $$T^{16}$$
$7$ $$5764801 - 67228 T^{4} + 646 T^{8} - 28 T^{12} + T^{16}$$
$11$ $$( -4 + T + T^{2} )^{8}$$
$13$ $$( 16384 + 273 T^{4} + T^{8} )^{2}$$
$17$ $$( 16384 + 273 T^{4} + T^{8} )^{2}$$
$19$ $$( 512 - 80 T^{2} + T^{4} )^{4}$$
$23$ $$( 4096 + 2576 T^{4} + T^{8} )^{2}$$
$29$ $$( 1444 + 77 T^{2} + T^{4} )^{4}$$
$31$ $$( 2048 + 112 T^{2} + T^{4} )^{4}$$
$37$ $$( 256 + T^{4} )^{4}$$
$41$ $$( 512 + 80 T^{2} + T^{4} )^{4}$$
$43$ $$( 4096 + 6928 T^{4} + T^{8} )^{2}$$
$47$ $$( 64 + 33 T^{4} + T^{8} )^{2}$$
$53$ $$( 16777216 + 12544 T^{4} + T^{8} )^{2}$$
$59$ $$( 2048 - 160 T^{2} + T^{4} )^{4}$$
$61$ $$( 2048 + 112 T^{2} + T^{4} )^{4}$$
$67$ $$( 65536 + 784 T^{4} + T^{8} )^{2}$$
$71$ $$( -64 + 4 T + T^{2} )^{8}$$
$73$ $$( 4194304 + 8448 T^{4} + T^{8} )^{2}$$
$79$ $$( 10816 + 217 T^{2} + T^{4} )^{4}$$
$83$ $$( 133448704 + 36432 T^{4} + T^{8} )^{2}$$
$89$ $$( 2048 - 112 T^{2} + T^{4} )^{4}$$
$97$ $$( 467943424 + 128961 T^{4} + T^{8} )^{2}$$