# Properties

 Label 700.2.c.h Level $700$ Weight $2$ Character orbit 700.c Analytic conductor $5.590$ Analytic rank $0$ Dimension $8$ CM discriminant -7 Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 700.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.58952814149$$ Analytic rank: $$0$$ Dimension: $$8$$ Coefficient field: 8.0.49787136.1 Defining polynomial: $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}\cdot 5^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{7} ) q^{7} + ( -\beta_{1} + \beta_{3} ) q^{8} + 3 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} + ( \beta_{1} - \beta_{7} ) q^{7} + ( -\beta_{1} + \beta_{3} ) q^{8} + 3 q^{9} + ( 1 - 2 \beta_{6} ) q^{11} + ( 1 + \beta_{2} + \beta_{6} ) q^{14} + ( -\beta_{2} + \beta_{4} ) q^{16} + 3 \beta_{1} q^{18} + ( \beta_{1} + 2 \beta_{5} - 2 \beta_{7} ) q^{22} + ( \beta_{1} + 4 \beta_{5} - \beta_{7} ) q^{23} + ( \beta_{1} + \beta_{3} - \beta_{5} + \beta_{7} ) q^{28} + ( -1 + 4 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} ) q^{29} + ( \beta_{1} - \beta_{3} + 3 \beta_{5} + \beta_{7} ) q^{32} + ( -3 + 3 \beta_{2} ) q^{36} + ( 5 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{37} + ( -\beta_{1} + 6 \beta_{5} + \beta_{7} ) q^{43} + ( 1 - \beta_{2} + 4 \beta_{6} ) q^{44} + ( -3 - 3 \beta_{2} + 5 \beta_{6} ) q^{46} + 7 q^{49} + ( 2 \beta_{1} - 4 \beta_{3} + 2 \beta_{7} ) q^{53} + ( -3 + 2 \beta_{2} + \beta_{4} - 2 \beta_{6} ) q^{56} + ( -3 \beta_{1} + 4 \beta_{3} - 4 \beta_{5} - 4 \beta_{7} ) q^{58} + ( 3 \beta_{1} - 3 \beta_{7} ) q^{63} + ( -5 - 2 \beta_{2} - \beta_{4} + 2 \beta_{6} ) q^{64} + ( -3 \beta_{1} - 2 \beta_{5} + 3 \beta_{7} ) q^{67} + ( 3 - 4 \beta_{2} - 2 \beta_{4} - 4 \beta_{6} ) q^{71} + ( -3 \beta_{1} + 3 \beta_{3} ) q^{72} + ( -11 + 3 \beta_{2} + 2 \beta_{4} + \beta_{6} ) q^{74} + ( -7 \beta_{1} + 2 \beta_{3} - 2 \beta_{5} - 3 \beta_{7} ) q^{77} + ( 1 - 4 \beta_{2} - 2 \beta_{4} ) q^{79} + 9 q^{81} + ( -7 - 7 \beta_{2} + 5 \beta_{6} ) q^{86} + ( \beta_{1} - \beta_{3} - 4 \beta_{5} + 4 \beta_{7} ) q^{88} + ( -3 \beta_{1} - 3 \beta_{3} - 5 \beta_{5} + 5 \beta_{7} ) q^{92} + 7 \beta_{1} q^{98} + ( 3 - 6 \beta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$8q - 6q^{4} + 24q^{9} + O(q^{10})$$ $$8q - 6q^{4} + 24q^{9} + 14q^{14} - 2q^{16} - 8q^{29} - 18q^{36} + 22q^{44} - 10q^{46} + 56q^{49} - 28q^{56} - 36q^{64} - 78q^{74} + 72q^{81} - 50q^{86} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} + 3 x^{6} + 5 x^{4} + 12 x^{2} + 16$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} + \nu^{2} + 1$$ $$\beta_{5}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{5} - \nu^{3} - 8 \nu$$$$)/8$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{6} + \nu^{4} + 3 \nu^{2} + 8$$$$)/2$$ $$\beta_{7}$$ $$=$$ $$($$$$3 \nu^{7} + 5 \nu^{5} + 11 \nu^{3} + 24 \nu$$$$)/8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} - \beta_{2}$$ $$\nu^{5}$$ $$=$$ $$\beta_{7} + 3 \beta_{5} - \beta_{3} + \beta_{1}$$ $$\nu^{6}$$ $$=$$ $$2 \beta_{6} - \beta_{4} - 2 \beta_{2} - 5$$ $$\nu^{7}$$ $$=$$ $$\beta_{7} - 5 \beta_{5} - 2 \beta_{3} - 6 \beta_{1}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/700\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$351$$ $$477$$ $$\chi(n)$$ $$-1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
699.1
 −1.09445 − 0.895644i −1.09445 + 0.895644i −0.228425 − 1.39564i −0.228425 + 1.39564i 0.228425 − 1.39564i 0.228425 + 1.39564i 1.09445 − 0.895644i 1.09445 + 0.895644i
−1.09445 0.895644i 0 0.395644 + 1.96048i 0 0 −2.64575 1.32288 2.50000i 3.00000 0
699.2 −1.09445 + 0.895644i 0 0.395644 1.96048i 0 0 −2.64575 1.32288 + 2.50000i 3.00000 0
699.3 −0.228425 1.39564i 0 −1.89564 + 0.637600i 0 0 −2.64575 1.32288 + 2.50000i 3.00000 0
699.4 −0.228425 + 1.39564i 0 −1.89564 0.637600i 0 0 −2.64575 1.32288 2.50000i 3.00000 0
699.5 0.228425 1.39564i 0 −1.89564 0.637600i 0 0 2.64575 −1.32288 + 2.50000i 3.00000 0
699.6 0.228425 + 1.39564i 0 −1.89564 + 0.637600i 0 0 2.64575 −1.32288 2.50000i 3.00000 0
699.7 1.09445 0.895644i 0 0.395644 1.96048i 0 0 2.64575 −1.32288 2.50000i 3.00000 0
699.8 1.09445 + 0.895644i 0 0.395644 + 1.96048i 0 0 2.64575 −1.32288 + 2.50000i 3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 699.8 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 CM by $$\Q(\sqrt{-7})$$
4.b odd 2 1 inner
5.b even 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.c.h 8
4.b odd 2 1 inner 700.2.c.h 8
5.b even 2 1 inner 700.2.c.h 8
5.c odd 4 1 700.2.g.c 4
5.c odd 4 1 700.2.g.e yes 4
7.b odd 2 1 CM 700.2.c.h 8
20.d odd 2 1 inner 700.2.c.h 8
20.e even 4 1 700.2.g.c 4
20.e even 4 1 700.2.g.e yes 4
28.d even 2 1 inner 700.2.c.h 8
35.c odd 2 1 inner 700.2.c.h 8
35.f even 4 1 700.2.g.c 4
35.f even 4 1 700.2.g.e yes 4
140.c even 2 1 inner 700.2.c.h 8
140.j odd 4 1 700.2.g.c 4
140.j odd 4 1 700.2.g.e yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.c.h 8 1.a even 1 1 trivial
700.2.c.h 8 4.b odd 2 1 inner
700.2.c.h 8 5.b even 2 1 inner
700.2.c.h 8 7.b odd 2 1 CM
700.2.c.h 8 20.d odd 2 1 inner
700.2.c.h 8 28.d even 2 1 inner
700.2.c.h 8 35.c odd 2 1 inner
700.2.c.h 8 140.c even 2 1 inner
700.2.g.c 4 5.c odd 4 1
700.2.g.c 4 20.e even 4 1
700.2.g.c 4 35.f even 4 1
700.2.g.c 4 140.j odd 4 1
700.2.g.e yes 4 5.c odd 4 1
700.2.g.e yes 4 20.e even 4 1
700.2.g.e yes 4 35.f even 4 1
700.2.g.e yes 4 140.j odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(700, [\chi])$$:

 $$T_{3}$$ $$T_{11}^{4} + 38 T_{11}^{2} + 25$$ $$T_{13}$$ $$T_{19}$$ $$T_{23}^{4} - 110 T_{23}^{2} + 1681$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + 12 T^{2} + 5 T^{4} + 3 T^{6} + T^{8}$$
$3$ $$T^{8}$$
$5$ $$T^{8}$$
$7$ $$( -7 + T^{2} )^{4}$$
$11$ $$( 25 + 38 T^{2} + T^{4} )^{2}$$
$13$ $$T^{8}$$
$17$ $$T^{8}$$
$19$ $$T^{8}$$
$23$ $$( 1681 - 110 T^{2} + T^{4} )^{2}$$
$29$ $$( -83 + 2 T + T^{2} )^{4}$$
$31$ $$T^{8}$$
$37$ $$( 5625 + 186 T^{2} + T^{4} )^{2}$$
$41$ $$T^{8}$$
$43$ $$( 10201 - 230 T^{2} + T^{4} )^{2}$$
$47$ $$T^{8}$$
$53$ $$( 100 + T^{2} )^{4}$$
$59$ $$T^{8}$$
$61$ $$T^{8}$$
$67$ $$( 2601 - 150 T^{2} + T^{4} )^{2}$$
$71$ $$( 34225 + 398 T^{2} + T^{4} )^{2}$$
$73$ $$T^{8}$$
$79$ $$( 225 + 222 T^{2} + T^{4} )^{2}$$
$83$ $$T^{8}$$
$89$ $$T^{8}$$
$97$ $$T^{8}$$