# Properties

 Label 700.2.i.e Level $700$ Weight $2$ Character orbit 700.i Analytic conductor $5.590$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 700.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.58952814149$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{3})$$ Coefficient field: 6.0.1783323.2 Defining polynomial: $$x^{6} - x^{5} + 5 x^{4} - 2 x^{3} + 19 x^{2} - 12 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$3$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + ( \beta_{3} - \beta_{5} ) q^{3} + ( -1 + \beta_{2} + \beta_{3} - \beta_{5} ) q^{7} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{3} - \beta_{5} ) q^{3} + ( -1 + \beta_{2} + \beta_{3} - \beta_{5} ) q^{7} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + 3 \beta_{4} - 2 \beta_{5} ) q^{9} + ( -1 - \beta_{3} - \beta_{4} + \beta_{5} ) q^{11} + ( -3 + \beta_{3} ) q^{13} + ( 4 + 2 \beta_{1} + \beta_{2} + 4 \beta_{4} - \beta_{5} ) q^{17} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{5} ) q^{19} + ( \beta_{2} - 2 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{21} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{23} + ( -3 - \beta_{1} + \beta_{2} - 2 \beta_{3} ) q^{27} + ( -\beta_{1} + \beta_{2} - \beta_{3} ) q^{29} + ( 1 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{4} - 3 \beta_{5} ) q^{31} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} - 6 \beta_{4} + 3 \beta_{5} ) q^{33} + ( -\beta_{4} + 3 \beta_{5} ) q^{37} + ( 6 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 6 \beta_{4} + \beta_{5} ) q^{39} + ( 2 - \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{41} + ( 4 + \beta_{1} - \beta_{2} ) q^{43} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{47} + ( 1 + \beta_{1} + \beta_{2} - 2 \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{49} + ( -3 \beta_{4} - 6 \beta_{5} ) q^{51} + ( 5 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} + 5 \beta_{4} + 4 \beta_{5} ) q^{53} + ( 3 - 2 \beta_{3} ) q^{57} + ( 2 - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{59} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} + 7 \beta_{4} + \beta_{5} ) q^{61} + ( -6 - \beta_{1} - \beta_{2} - 4 \beta_{3} - 12 \beta_{4} + 3 \beta_{5} ) q^{63} + ( 6 + 2 \beta_{3} + 6 \beta_{4} - 2 \beta_{5} ) q^{67} + ( -9 - \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{69} + ( -7 + 2 \beta_{3} ) q^{71} + ( 4 + 4 \beta_{4} ) q^{73} + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{77} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{79} + ( -6 - 5 \beta_{3} - 6 \beta_{4} + 5 \beta_{5} ) q^{81} + ( -10 + \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{83} + ( -9 - 4 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} - 9 \beta_{4} + 6 \beta_{5} ) q^{87} + ( -\beta_{1} - 2 \beta_{2} - \beta_{3} - 5 \beta_{4} + 2 \beta_{5} ) q^{89} + ( 6 + \beta_{1} - 2 \beta_{2} - 4 \beta_{3} + 3 \beta_{4} + \beta_{5} ) q^{91} + ( 2 \beta_{1} + 4 \beta_{2} + 2 \beta_{3} + 9 \beta_{4} - 7 \beta_{5} ) q^{93} + ( -4 + 2 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{97} + ( 6 + 2 \beta_{1} - 2 \beta_{2} + 5 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q + q^{3} - 4q^{7} - 8q^{9} + O(q^{10})$$ $$6q + q^{3} - 4q^{7} - 8q^{9} - 4q^{11} - 16q^{13} + 10q^{17} + 2q^{19} - 11q^{21} - 3q^{23} - 20q^{27} + 3q^{31} + 18q^{33} + 6q^{37} + 14q^{39} + 22q^{41} + 22q^{43} + 4q^{47} + 12q^{49} + 3q^{51} + 14q^{53} + 14q^{57} + 5q^{59} - 17q^{61} - 5q^{63} + 20q^{67} - 48q^{69} - 38q^{71} + 12q^{73} + 13q^{77} + q^{79} - 23q^{81} - 56q^{83} - 27q^{87} + 14q^{89} + 17q^{91} - 28q^{93} - 30q^{97} + 42q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 5 x^{4} - 2 x^{3} + 19 x^{2} - 12 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-3 \nu^{5} + 15 \nu^{4} + 8 \nu^{3} + 57 \nu^{2} + 47 \nu + 180$$$$)/83$$ $$\beta_{2}$$ $$=$$ $$($$$$4 \nu^{5} - 20 \nu^{4} + 17 \nu^{3} - 76 \nu^{2} + 131 \nu - 240$$$$)/83$$ $$\beta_{3}$$ $$=$$ $$($$$$-5 \nu^{5} + 25 \nu^{4} - 42 \nu^{3} + 95 \nu^{2} - 60 \nu + 300$$$$)/83$$ $$\beta_{4}$$ $$=$$ $$($$$$-20 \nu^{5} + 17 \nu^{4} - 85 \nu^{3} - 35 \nu^{2} - 323 \nu - 45$$$$)/249$$ $$\beta_{5}$$ $$=$$ $$($$$$16 \nu^{5} + 3 \nu^{4} + 68 \nu^{3} + 28 \nu^{2} + 358 \nu + 36$$$$)/83$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{2} + \beta_{1}$$$$)/3$$ $$\nu^{2}$$ $$=$$ $$($$$$-3 \beta_{5} - 9 \beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_{1} - 9$$$$)/3$$ $$\nu^{3}$$ $$=$$ $$($$$$-7 \beta_{3} - 5 \beta_{2} + 5 \beta_{1}$$$$)/3$$ $$\nu^{4}$$ $$=$$ $$5 \beta_{5} + 12 \beta_{4} - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_{1}$$ $$\nu^{5}$$ $$=$$ $$($$$$18 \beta_{5} + 9 \beta_{4} + 5 \beta_{3} - 23 \beta_{2} - 46 \beta_{1} + 9$$$$)/3$$

## Character values

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

 $$n$$ $$101$$ $$351$$ $$477$$ $$\chi(n)$$ $$\beta_{4}$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
401.1
 −0.956115 + 1.65604i 1.09935 − 1.90412i 0.356769 − 0.617942i −0.956115 − 1.65604i 1.09935 + 1.90412i 0.356769 + 0.617942i
0 −1.28442 2.22469i 0 0 0 −2.58392 0.568650i 0 −1.79949 + 3.11682i 0
401.2 0 0.182224 + 0.315621i 0 0 0 2.11581 1.58850i 0 1.43359 2.48305i 0
401.3 0 1.60220 + 2.77509i 0 0 0 −1.53189 + 2.15715i 0 −3.63409 + 6.29444i 0
501.1 0 −1.28442 + 2.22469i 0 0 0 −2.58392 + 0.568650i 0 −1.79949 3.11682i 0
501.2 0 0.182224 0.315621i 0 0 0 2.11581 + 1.58850i 0 1.43359 + 2.48305i 0
501.3 0 1.60220 2.77509i 0 0 0 −1.53189 2.15715i 0 −3.63409 6.29444i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 501.3 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 700.2.i.e yes 6
5.b even 2 1 700.2.i.d 6
5.c odd 4 2 700.2.r.d 12
7.c even 3 1 inner 700.2.i.e yes 6
7.c even 3 1 4900.2.a.ba 3
7.d odd 6 1 4900.2.a.bd 3
35.i odd 6 1 4900.2.a.bb 3
35.j even 6 1 700.2.i.d 6
35.j even 6 1 4900.2.a.bc 3
35.k even 12 2 4900.2.e.s 6
35.l odd 12 2 700.2.r.d 12
35.l odd 12 2 4900.2.e.t 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
700.2.i.d 6 5.b even 2 1
700.2.i.d 6 35.j even 6 1
700.2.i.e yes 6 1.a even 1 1 trivial
700.2.i.e yes 6 7.c even 3 1 inner
700.2.r.d 12 5.c odd 4 2
700.2.r.d 12 35.l odd 12 2
4900.2.a.ba 3 7.c even 3 1
4900.2.a.bb 3 35.i odd 6 1
4900.2.a.bc 3 35.j even 6 1
4900.2.a.bd 3 7.d odd 6 1
4900.2.e.s 6 35.k even 12 2
4900.2.e.t 6 35.l odd 12 2

## Hecke kernels

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

 $$T_{3}^{6} - T_{3}^{5} + 9 T_{3}^{4} + 2 T_{3}^{3} + 67 T_{3}^{2} - 24 T_{3} + 9$$ $$T_{11}^{6} + 4 T_{11}^{5} + 19 T_{11}^{4} + 6 T_{11}^{3} + 45 T_{11}^{2} + 27 T_{11} + 81$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{6}$$
$3$ $$9 - 24 T + 67 T^{2} + 2 T^{3} + 9 T^{4} - T^{5} + T^{6}$$
$5$ $$T^{6}$$
$7$ $$343 + 196 T + 14 T^{2} - 11 T^{3} + 2 T^{4} + 4 T^{5} + T^{6}$$
$11$ $$81 + 27 T + 45 T^{2} + 6 T^{3} + 19 T^{4} + 4 T^{5} + T^{6}$$
$13$ $$( -3 + 13 T + 8 T^{2} + T^{3} )^{2}$$
$17$ $$6561 + 729 T + 891 T^{2} - 252 T^{3} + 91 T^{4} - 10 T^{5} + T^{6}$$
$19$ $$441 + 483 T + 487 T^{2} + 88 T^{3} + 27 T^{4} - 2 T^{5} + T^{6}$$
$23$ $$6561 + 2916 T + 1539 T^{2} + 54 T^{3} + 45 T^{4} + 3 T^{5} + T^{6}$$
$29$ $$( 81 - 45 T + T^{3} )^{2}$$
$31$ $$1369 + 1554 T + 1653 T^{2} + 200 T^{3} + 51 T^{4} - 3 T^{5} + T^{6}$$
$37$ $$22201 - 9387 T + 4863 T^{2} + 80 T^{3} + 99 T^{4} - 6 T^{5} + T^{6}$$
$41$ $$( 873 - 78 T - 11 T^{2} + T^{3} )^{2}$$
$43$ $$( 71 + 14 T - 11 T^{2} + T^{3} )^{2}$$
$47$ $$81 - 189 T + 477 T^{2} + 66 T^{3} + 37 T^{4} - 4 T^{5} + T^{6}$$
$53$ $$301401 - 8235 T + 7911 T^{2} - 888 T^{3} + 211 T^{4} - 14 T^{5} + T^{6}$$
$59$ $$81 + 45 T^{2} - 18 T^{3} + 25 T^{4} - 5 T^{5} + T^{6}$$
$61$ $$1 + 26 T + 659 T^{2} + 440 T^{3} + 263 T^{4} + 17 T^{5} + T^{6}$$
$67$ $$5184 - 7200 T + 8560 T^{2} - 1856 T^{3} + 300 T^{4} - 20 T^{5} + T^{6}$$
$71$ $$( 45 + 87 T + 19 T^{2} + T^{3} )^{2}$$
$73$ $$( 16 - 4 T + T^{2} )^{3}$$
$79$ $$201601 + 52982 T + 13475 T^{2} + 1016 T^{3} + 119 T^{4} - T^{5} + T^{6}$$
$83$ $$( -981 + 129 T + 28 T^{2} + T^{3} )^{2}$$
$89$ $$2025 + 1755 T + 2151 T^{2} - 636 T^{3} + 157 T^{4} - 14 T^{5} + T^{6}$$
$97$ $$( -5 - 18 T + 15 T^{2} + T^{3} )^{2}$$