# Properties

 Label 700.2.t.b Level $700$ Weight $2$ Character orbit 700.t Analytic conductor $5.590$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 700.t (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.58952814149$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 28) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{2} + (\zeta_{12}^{2} + 1) q^{3} + 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{6} + (2 \zeta_{12}^{2} + 1) q^{7} + (2 \zeta_{12}^{3} + 2) q^{8}+O(q^{10})$$ q + (-z^3 + z^2 + z) * q^2 + (z^2 + 1) * q^3 + 2*z * q^4 + (-z^3 + 2*z^2 + 2*z - 1) * q^6 + (2*z^2 + 1) * q^7 + (2*z^3 + 2) * q^8 $$q + ( - \zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}) q^{2} + (\zeta_{12}^{2} + 1) q^{3} + 2 \zeta_{12} q^{4} + ( - \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 1) q^{6} + (2 \zeta_{12}^{2} + 1) q^{7} + (2 \zeta_{12}^{3} + 2) q^{8} - \zeta_{12} q^{11} + (2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{12} + (2 \zeta_{12}^{3} - 4 \zeta_{12}) q^{13} + ( - \zeta_{12}^{3} + 3 \zeta_{12}^{2} + 3 \zeta_{12} - 2) q^{14} + 4 \zeta_{12}^{2} q^{16} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{17} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{19} + (5 \zeta_{12}^{2} - 1) q^{21} + ( - \zeta_{12}^{3} - 1) q^{22} + \zeta_{12}^{2} q^{23} + (4 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{24} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 2 \zeta_{12} - 4) q^{26} + ( - 6 \zeta_{12}^{2} + 3) q^{27} + (4 \zeta_{12}^{3} + 2 \zeta_{12}) q^{28} - 4 q^{29} + ( - 2 \zeta_{12}^{3} + \zeta_{12}) q^{31} + (4 \zeta_{12}^{2} + 4 \zeta_{12} - 4) q^{32} + ( - \zeta_{12}^{3} - \zeta_{12}) q^{33} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{34} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{37} + ( - 6 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} - 3) q^{38} - 6 \zeta_{12} q^{39} + (4 \zeta_{12}^{2} - 2) q^{41} + (\zeta_{12}^{3} + 4 \zeta_{12}^{2} + 4 \zeta_{12} - 5) q^{42} - 2 q^{43} - 2 \zeta_{12}^{2} q^{44} + (\zeta_{12}^{2} + \zeta_{12} - 1) q^{46} + ( - 5 \zeta_{12}^{2} + 10) q^{47} + (8 \zeta_{12}^{2} - 4) q^{48} + (8 \zeta_{12}^{2} - 3) q^{49} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{51} + ( - 4 \zeta_{12}^{2} - 4) q^{52} - \zeta_{12} q^{53} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12} + 6) q^{54} + (6 \zeta_{12}^{3} + 4 \zeta_{12}^{2} - 4 \zeta_{12} + 2) q^{56} - 9 \zeta_{12}^{3} q^{57} + (4 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 4 \zeta_{12}) q^{58} + (6 \zeta_{12}^{3} - 3 \zeta_{12}) q^{59} + (3 \zeta_{12}^{2} - 6) q^{61} + ( - \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 2 \zeta_{12} + 1) q^{62} + 8 \zeta_{12}^{3} q^{64} + ( - 2 \zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12} - 1) q^{66} + ( - 3 \zeta_{12}^{2} + 3) q^{67} + ( - 2 \zeta_{12}^{2} + 4) q^{68} + (2 \zeta_{12}^{2} - 1) q^{69} + 14 \zeta_{12}^{3} q^{71} + ( - 10 \zeta_{12}^{3} + 5 \zeta_{12}) q^{73} + ( - 3 \zeta_{12}^{2} + 3 \zeta_{12} + 3) q^{74} + ( - 12 \zeta_{12}^{2} + 6) q^{76} + ( - 2 \zeta_{12}^{3} - \zeta_{12}) q^{77} + ( - 6 \zeta_{12}^{3} - 6) q^{78} + ( - 9 \zeta_{12}^{3} + 9 \zeta_{12}) q^{79} + ( - 9 \zeta_{12}^{2} + 9) q^{81} + (2 \zeta_{12}^{3} + 2 \zeta_{12}^{2} + 2 \zeta_{12} - 4) q^{82} + ( - 16 \zeta_{12}^{2} + 8) q^{83} + (10 \zeta_{12}^{3} - 2 \zeta_{12}) q^{84} + (2 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 2 \zeta_{12}) q^{86} + ( - 4 \zeta_{12}^{2} - 4) q^{87} + ( - 2 \zeta_{12}^{2} - 2 \zeta_{12} + 2) q^{88} + (9 \zeta_{12}^{2} - 18) q^{89} + ( - 2 \zeta_{12}^{3} - 8 \zeta_{12}) q^{91} + 2 \zeta_{12}^{3} q^{92} + ( - 3 \zeta_{12}^{3} + 3 \zeta_{12}) q^{93} + ( - 10 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12} + 5) q^{94} + (4 \zeta_{12}^{3} + 4 \zeta_{12}^{2} + 4 \zeta_{12} - 8) q^{96} + (10 \zeta_{12}^{3} - 20 \zeta_{12}) q^{97} + (3 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 5 \zeta_{12} - 8) q^{98} +O(q^{100})$$ q + (-z^3 + z^2 + z) * q^2 + (z^2 + 1) * q^3 + 2*z * q^4 + (-z^3 + 2*z^2 + 2*z - 1) * q^6 + (2*z^2 + 1) * q^7 + (2*z^3 + 2) * q^8 - z * q^11 + (2*z^3 + 2*z) * q^12 + (2*z^3 - 4*z) * q^13 + (-z^3 + 3*z^2 + 3*z - 2) * q^14 + 4*z^2 * q^16 + (-2*z^3 + z) * q^17 + (-3*z^3 - 3*z) * q^19 + (5*z^2 - 1) * q^21 + (-z^3 - 1) * q^22 + z^2 * q^23 + (4*z^3 + 2*z^2 - 2*z + 2) * q^24 + (-2*z^3 + 2*z^2 - 2*z - 4) * q^26 + (-6*z^2 + 3) * q^27 + (4*z^3 + 2*z) * q^28 - 4 * q^29 + (-2*z^3 + z) * q^31 + (4*z^2 + 4*z - 4) * q^32 + (-z^3 - z) * q^33 + (-z^3 - 2*z^2 + 2*z + 1) * q^34 + (-3*z^3 + 3*z) * q^37 + (-6*z^3 - 3*z^2 + 3*z - 3) * q^38 - 6*z * q^39 + (4*z^2 - 2) * q^41 + (z^3 + 4*z^2 + 4*z - 5) * q^42 - 2 * q^43 - 2*z^2 * q^44 + (z^2 + z - 1) * q^46 + (-5*z^2 + 10) * q^47 + (8*z^2 - 4) * q^48 + (8*z^2 - 3) * q^49 + (-3*z^3 + 3*z) * q^51 + (-4*z^2 - 4) * q^52 - z * q^53 + (-3*z^3 - 3*z^2 - 3*z + 6) * q^54 + (6*z^3 + 4*z^2 - 4*z + 2) * q^56 - 9*z^3 * q^57 + (4*z^3 - 4*z^2 - 4*z) * q^58 + (6*z^3 - 3*z) * q^59 + (3*z^2 - 6) * q^61 + (-z^3 - 2*z^2 + 2*z + 1) * q^62 + 8*z^3 * q^64 + (-2*z^3 - z^2 + z - 1) * q^66 + (-3*z^2 + 3) * q^67 + (-2*z^2 + 4) * q^68 + (2*z^2 - 1) * q^69 + 14*z^3 * q^71 + (-10*z^3 + 5*z) * q^73 + (-3*z^2 + 3*z + 3) * q^74 + (-12*z^2 + 6) * q^76 + (-2*z^3 - z) * q^77 + (-6*z^3 - 6) * q^78 + (-9*z^3 + 9*z) * q^79 + (-9*z^2 + 9) * q^81 + (2*z^3 + 2*z^2 + 2*z - 4) * q^82 + (-16*z^2 + 8) * q^83 + (10*z^3 - 2*z) * q^84 + (2*z^3 - 2*z^2 - 2*z) * q^86 + (-4*z^2 - 4) * q^87 + (-2*z^2 - 2*z + 2) * q^88 + (9*z^2 - 18) * q^89 + (-2*z^3 - 8*z) * q^91 + 2*z^3 * q^92 + (-3*z^3 + 3*z) * q^93 + (-10*z^3 + 5*z^2 + 5*z + 5) * q^94 + (4*z^3 + 4*z^2 + 4*z - 8) * q^96 + (10*z^3 - 20*z) * q^97 + (3*z^3 + 5*z^2 + 5*z - 8) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{2} + 6 q^{3} + 8 q^{7} + 8 q^{8}+O(q^{10})$$ 4 * q + 2 * q^2 + 6 * q^3 + 8 * q^7 + 8 * q^8 $$4 q + 2 q^{2} + 6 q^{3} + 8 q^{7} + 8 q^{8} - 2 q^{14} + 8 q^{16} + 6 q^{21} - 4 q^{22} + 2 q^{23} + 12 q^{24} - 12 q^{26} - 16 q^{29} - 8 q^{32} - 18 q^{38} - 12 q^{42} - 8 q^{43} - 4 q^{44} - 2 q^{46} + 30 q^{47} + 4 q^{49} - 24 q^{52} + 18 q^{54} + 16 q^{56} - 8 q^{58} - 18 q^{61} - 6 q^{66} + 6 q^{67} + 12 q^{68} + 6 q^{74} - 24 q^{78} + 18 q^{81} - 12 q^{82} - 4 q^{86} - 24 q^{87} + 4 q^{88} - 54 q^{89} + 30 q^{94} - 24 q^{96} - 22 q^{98}+O(q^{100})$$ 4 * q + 2 * q^2 + 6 * q^3 + 8 * q^7 + 8 * q^8 - 2 * q^14 + 8 * q^16 + 6 * q^21 - 4 * q^22 + 2 * q^23 + 12 * q^24 - 12 * q^26 - 16 * q^29 - 8 * q^32 - 18 * q^38 - 12 * q^42 - 8 * q^43 - 4 * q^44 - 2 * q^46 + 30 * q^47 + 4 * q^49 - 24 * q^52 + 18 * q^54 + 16 * q^56 - 8 * q^58 - 18 * q^61 - 6 * q^66 + 6 * q^67 + 12 * q^68 + 6 * q^74 - 24 * q^78 + 18 * q^81 - 12 * q^82 - 4 * q^86 - 24 * q^87 + 4 * q^88 - 54 * q^89 + 30 * q^94 - 24 * q^96 - 22 * q^98

## Character values

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

 $$n$$ $$101$$ $$351$$ $$477$$ $$\chi(n)$$ $$\zeta_{12}^{2}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
−0.366025 + 1.36603i 1.50000 + 0.866025i −1.73205 1.00000i 0 −1.73205 + 1.73205i 2.00000 + 1.73205i 2.00000 2.00000i 0 0
199.2 1.36603 + 0.366025i 1.50000 + 0.866025i 1.73205 + 1.00000i 0 1.73205 + 1.73205i 2.00000 + 1.73205i 2.00000 + 2.00000i 0 0
299.1 −0.366025 1.36603i 1.50000 0.866025i −1.73205 + 1.00000i 0 −1.73205 1.73205i 2.00000 1.73205i 2.00000 + 2.00000i 0 0
299.2 1.36603 0.366025i 1.50000 0.866025i 1.73205 1.00000i 0 1.73205 1.73205i 2.00000 1.73205i 2.00000 2.00000i 0 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
7.d odd 6 1 inner
20.d odd 2 1 inner
140.s even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 700.2.t.b 4
4.b odd 2 1 700.2.t.a 4
5.b even 2 1 700.2.t.a 4
5.c odd 4 1 28.2.f.a 4
5.c odd 4 1 700.2.p.a 4
7.d odd 6 1 inner 700.2.t.b 4
15.e even 4 1 252.2.bf.e 4
20.d odd 2 1 inner 700.2.t.b 4
20.e even 4 1 28.2.f.a 4
20.e even 4 1 700.2.p.a 4
28.f even 6 1 700.2.t.a 4
35.f even 4 1 196.2.f.a 4
35.i odd 6 1 700.2.t.a 4
35.k even 12 1 28.2.f.a 4
35.k even 12 1 196.2.d.b 4
35.k even 12 1 700.2.p.a 4
35.l odd 12 1 196.2.d.b 4
35.l odd 12 1 196.2.f.a 4
40.i odd 4 1 448.2.p.d 4
40.k even 4 1 448.2.p.d 4
60.l odd 4 1 252.2.bf.e 4
105.w odd 12 1 252.2.bf.e 4
105.w odd 12 1 1764.2.b.a 4
105.x even 12 1 1764.2.b.a 4
140.j odd 4 1 196.2.f.a 4
140.s even 6 1 inner 700.2.t.b 4
140.w even 12 1 196.2.d.b 4
140.w even 12 1 196.2.f.a 4
140.x odd 12 1 28.2.f.a 4
140.x odd 12 1 196.2.d.b 4
140.x odd 12 1 700.2.p.a 4
280.bp odd 12 1 448.2.p.d 4
280.bp odd 12 1 3136.2.f.e 4
280.br even 12 1 3136.2.f.e 4
280.bt odd 12 1 3136.2.f.e 4
280.bv even 12 1 448.2.p.d 4
280.bv even 12 1 3136.2.f.e 4
420.bp odd 12 1 1764.2.b.a 4
420.br even 12 1 252.2.bf.e 4
420.br even 12 1 1764.2.b.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
28.2.f.a 4 5.c odd 4 1
28.2.f.a 4 20.e even 4 1
28.2.f.a 4 35.k even 12 1
28.2.f.a 4 140.x odd 12 1
196.2.d.b 4 35.k even 12 1
196.2.d.b 4 35.l odd 12 1
196.2.d.b 4 140.w even 12 1
196.2.d.b 4 140.x odd 12 1
196.2.f.a 4 35.f even 4 1
196.2.f.a 4 35.l odd 12 1
196.2.f.a 4 140.j odd 4 1
196.2.f.a 4 140.w even 12 1
252.2.bf.e 4 15.e even 4 1
252.2.bf.e 4 60.l odd 4 1
252.2.bf.e 4 105.w odd 12 1
252.2.bf.e 4 420.br even 12 1
448.2.p.d 4 40.i odd 4 1
448.2.p.d 4 40.k even 4 1
448.2.p.d 4 280.bp odd 12 1
448.2.p.d 4 280.bv even 12 1
700.2.p.a 4 5.c odd 4 1
700.2.p.a 4 20.e even 4 1
700.2.p.a 4 35.k even 12 1
700.2.p.a 4 140.x odd 12 1
700.2.t.a 4 4.b odd 2 1
700.2.t.a 4 5.b even 2 1
700.2.t.a 4 28.f even 6 1
700.2.t.a 4 35.i odd 6 1
700.2.t.b 4 1.a even 1 1 trivial
700.2.t.b 4 7.d odd 6 1 inner
700.2.t.b 4 20.d odd 2 1 inner
700.2.t.b 4 140.s even 6 1 inner
1764.2.b.a 4 105.w odd 12 1
1764.2.b.a 4 105.x even 12 1
1764.2.b.a 4 420.bp odd 12 1
1764.2.b.a 4 420.br even 12 1
3136.2.f.e 4 280.bp odd 12 1
3136.2.f.e 4 280.br even 12 1
3136.2.f.e 4 280.bt odd 12 1
3136.2.f.e 4 280.bv even 12 1

## Hecke kernels

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

 $$T_{3}^{2} - 3T_{3} + 3$$ T3^2 - 3*T3 + 3 $$T_{13}^{2} - 12$$ T13^2 - 12

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4$$
$3$ $$(T^{2} - 3 T + 3)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} - 4 T + 7)^{2}$$
$11$ $$T^{4} - T^{2} + 1$$
$13$ $$(T^{2} - 12)^{2}$$
$17$ $$T^{4} + 3T^{2} + 9$$
$19$ $$T^{4} + 27T^{2} + 729$$
$23$ $$(T^{2} - T + 1)^{2}$$
$29$ $$(T + 4)^{4}$$
$31$ $$T^{4} + 3T^{2} + 9$$
$37$ $$T^{4} - 9T^{2} + 81$$
$41$ $$(T^{2} + 12)^{2}$$
$43$ $$(T + 2)^{4}$$
$47$ $$(T^{2} - 15 T + 75)^{2}$$
$53$ $$T^{4} - T^{2} + 1$$
$59$ $$T^{4} + 27T^{2} + 729$$
$61$ $$(T^{2} + 9 T + 27)^{2}$$
$67$ $$(T^{2} - 3 T + 9)^{2}$$
$71$ $$(T^{2} + 196)^{2}$$
$73$ $$T^{4} + 75T^{2} + 5625$$
$79$ $$T^{4} - 81T^{2} + 6561$$
$83$ $$(T^{2} + 192)^{2}$$
$89$ $$(T^{2} + 27 T + 243)^{2}$$
$97$ $$(T^{2} - 300)^{2}$$