# Properties

 Label 700.4.e.a Level $700$ Weight $4$ Character orbit 700.e Analytic conductor $41.301$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 700.e (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$41.3013370040$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 28) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 10 i q^{3} -7 i q^{7} -73 q^{9} +O(q^{10})$$ $$q + 10 i q^{3} -7 i q^{7} -73 q^{9} -40 q^{11} + 12 i q^{13} -58 i q^{17} -26 q^{19} + 70 q^{21} + 64 i q^{23} -460 i q^{27} + 62 q^{29} + 252 q^{31} -400 i q^{33} + 26 i q^{37} -120 q^{39} + 6 q^{41} -416 i q^{43} -396 i q^{47} -49 q^{49} + 580 q^{51} + 450 i q^{53} -260 i q^{57} -274 q^{59} -576 q^{61} + 511 i q^{63} -476 i q^{67} -640 q^{69} -448 q^{71} + 158 i q^{73} + 280 i q^{77} + 936 q^{79} + 2629 q^{81} -530 i q^{83} + 620 i q^{87} + 390 q^{89} + 84 q^{91} + 2520 i q^{93} + 214 i q^{97} + 2920 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 146q^{9} + O(q^{10})$$ $$2q - 146q^{9} - 80q^{11} - 52q^{19} + 140q^{21} + 124q^{29} + 504q^{31} - 240q^{39} + 12q^{41} - 98q^{49} + 1160q^{51} - 548q^{59} - 1152q^{61} - 1280q^{69} - 896q^{71} + 1872q^{79} + 5258q^{81} + 780q^{89} + 168q^{91} + 5840q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
449.1
 − 1.00000i 1.00000i
0 10.0000i 0 0 0 7.00000i 0 −73.0000 0
449.2 0 10.0000i 0 0 0 7.00000i 0 −73.0000 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.4.e.a 2
5.b even 2 1 inner 700.4.e.a 2
5.c odd 4 1 28.4.a.a 1
5.c odd 4 1 700.4.a.n 1
15.e even 4 1 252.4.a.d 1
20.e even 4 1 112.4.a.g 1
35.f even 4 1 196.4.a.d 1
35.k even 12 2 196.4.e.a 2
35.l odd 12 2 196.4.e.f 2
40.i odd 4 1 448.4.a.p 1
40.k even 4 1 448.4.a.a 1
60.l odd 4 1 1008.4.a.o 1
105.k odd 4 1 1764.4.a.c 1
105.w odd 12 2 1764.4.k.m 2
105.x even 12 2 1764.4.k.d 2
140.j odd 4 1 784.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.4.a.a 1 5.c odd 4 1
112.4.a.g 1 20.e even 4 1
196.4.a.d 1 35.f even 4 1
196.4.e.a 2 35.k even 12 2
196.4.e.f 2 35.l odd 12 2
252.4.a.d 1 15.e even 4 1
448.4.a.a 1 40.k even 4 1
448.4.a.p 1 40.i odd 4 1
700.4.a.n 1 5.c odd 4 1
700.4.e.a 2 1.a even 1 1 trivial
700.4.e.a 2 5.b even 2 1 inner
784.4.a.a 1 140.j odd 4 1
1008.4.a.o 1 60.l odd 4 1
1764.4.a.c 1 105.k odd 4 1
1764.4.k.d 2 105.x even 12 2
1764.4.k.m 2 105.w odd 12 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(700, [\chi])$$:

 $$T_{3}^{2} + 100$$ $$T_{11} + 40$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$100 + T^{2}$$
$5$ $$T^{2}$$
$7$ $$49 + T^{2}$$
$11$ $$( 40 + T )^{2}$$
$13$ $$144 + T^{2}$$
$17$ $$3364 + T^{2}$$
$19$ $$( 26 + T )^{2}$$
$23$ $$4096 + T^{2}$$
$29$ $$( -62 + T )^{2}$$
$31$ $$( -252 + T )^{2}$$
$37$ $$676 + T^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$173056 + T^{2}$$
$47$ $$156816 + T^{2}$$
$53$ $$202500 + T^{2}$$
$59$ $$( 274 + T )^{2}$$
$61$ $$( 576 + T )^{2}$$
$67$ $$226576 + T^{2}$$
$71$ $$( 448 + T )^{2}$$
$73$ $$24964 + T^{2}$$
$79$ $$( -936 + T )^{2}$$
$83$ $$280900 + T^{2}$$
$89$ $$( -390 + T )^{2}$$
$97$ $$45796 + T^{2}$$