# Properties

 Label 700.2.c.b Level $700$ Weight $2$ Character orbit 700.c Analytic conductor $5.590$ Analytic rank $1$ Dimension $4$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [700,2,Mod(699,700)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(700, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("700.699");

S:= CuspForms(chi, 2);

N := Newforms(S);

 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

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$5.58952814149$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 140) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - 1) q^{2} + ( - \beta_{3} + \beta_{2}) q^{3} + (\beta_{2} + \beta_1) q^{4} + (\beta_1 - 2) q^{6} + (\beta_{3} - \beta_{2} - 2) q^{7} + (\beta_{3} - \beta_1 - 1) q^{8}+O(q^{10})$$ q + (-b3 - 1) * q^2 + (-b3 + b2) * q^3 + (b2 + b1) * q^4 + (b1 - 2) * q^6 + (b3 - b2 - 2) * q^7 + (b3 - b1 - 1) * q^8 $$q + ( - \beta_{3} - 1) q^{2} + ( - \beta_{3} + \beta_{2}) q^{3} + (\beta_{2} + \beta_1) q^{4} + (\beta_1 - 2) q^{6} + (\beta_{3} - \beta_{2} - 2) q^{7} + (\beta_{3} - \beta_1 - 1) q^{8} + (2 \beta_{3} - \beta_{2} - \beta_1 + 1) q^{11} + (2 \beta_{3} + \beta_{2} - \beta_1 + 2) q^{12} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 2) q^{13} + (2 \beta_{3} - \beta_1 + 4) q^{14} + (2 \beta_{3} - 2 \beta_{2} + 2) q^{16} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 - 4) q^{17} - 6 q^{19} + (2 \beta_{3} - 2 \beta_{2} + 3) q^{21} + ( - 2 \beta_{2} - \beta_1 + 2) q^{22} + (\beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{23} + (\beta_{3} - 4 \beta_{2} - \beta_1 - 1) q^{24} + (5 \beta_{3} - 2 \beta_{2} - 2 \beta_1 + 1) q^{26} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{27} + ( - 2 \beta_{3} - 3 \beta_{2} - \beta_1 - 2) q^{28} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 3) q^{29} - 6 q^{31} + ( - 2 \beta_{3} - 2 \beta_1 + 2) q^{32} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 2) q^{33} + (\beta_{3} + 2 \beta_{2} + 2 \beta_1 + 5) q^{34} + ( - \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 3) q^{37} + (6 \beta_{3} + 6) q^{38} + (6 \beta_{3} - 3 \beta_{2} - 3 \beta_1 + 3) q^{39} + ( - 2 \beta_{3} + 2 \beta_{2}) q^{41} + ( - 3 \beta_{3} - 2 \beta_1 + 1) q^{42} + 2 q^{43} + ( - 4 \beta_{3} + \beta_{2} + \beta_1) q^{44} + (4 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{46} + ( - \beta_{3} + \beta_{2}) q^{47} + ( - 2 \beta_{3} + 2 \beta_{2} + 6) q^{48} + ( - 4 \beta_{3} + 4 \beta_{2} + 1) q^{49} + ( - 3 \beta_{2} + 3 \beta_1 - 3) q^{51} + (2 \beta_{3} - 5 \beta_{2} - 3 \beta_1 + 6) q^{52} + ( - \beta_{3} + \beta_1 - 1) q^{53} + (3 \beta_1 - 6) q^{54} + ( - 3 \beta_{3} + 4 \beta_{2} + 3 \beta_1 + 3) q^{56} + (6 \beta_{3} - 6 \beta_{2}) q^{57} + ( - 3 \beta_{3} + 4 \beta_{2} + 4 \beta_1 + 5) q^{58} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{59} + ( - \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 3) q^{61} + (6 \beta_{3} + 6) q^{62} + ( - 4 \beta_{3} + 4 \beta_1 - 4) q^{64} + ( - 5 \beta_{3} + 2 \beta_{2} + 2 \beta_1 - 1) q^{66} + (\beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{67} + ( - 2 \beta_{3} - \beta_{2} - 3 \beta_1 - 6) q^{68} + (5 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 3) q^{69} + (2 \beta_{2} - 2 \beta_1 + 2) q^{71} + ( - 2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 - 8) q^{73} + (6 \beta_{2} - 2 \beta_1 + 4) q^{74} + ( - 6 \beta_{2} - 6 \beta_1) q^{76} + ( - 3 \beta_{3} + 4 \beta_{2} + 3 \beta_1 - 4) q^{77} + ( - 6 \beta_{2} - 3 \beta_1 + 6) q^{78} + (2 \beta_{3} - 5 \beta_{2} + 3 \beta_1 - 3) q^{79} - 9 q^{81} + (2 \beta_1 - 4) q^{82} + ( - 4 \beta_{3} - 2 \beta_{2} + 6 \beta_1 - 6) q^{83} + ( - 4 \beta_{3} + \beta_{2} + 5 \beta_1 - 4) q^{84} + ( - 2 \beta_{3} - 2) q^{86} + ( - 5 \beta_{3} - \beta_{2} + 6 \beta_1 - 6) q^{87} + ( - 3 \beta_{3} + 4 \beta_{2} + 3 \beta_1 - 5) q^{88} + (5 \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 3) q^{89} + ( - 8 \beta_{3} - \beta_{2} + \beta_1 + 1) q^{91} + (2 \beta_{3} - 4 \beta_{2} - 2 \beta_1 + 6) q^{92} + (6 \beta_{3} - 6 \beta_{2}) q^{93} + (\beta_1 - 2) q^{94} + ( - 6 \beta_{3} + 2 \beta_1 - 10) q^{96} + ( - 3 \beta_{3} - 6 \beta_{2} - 3 \beta_1 - 6) q^{97} + ( - \beta_{3} + 4 \beta_1 - 9) q^{98}+O(q^{100})$$ q + (-b3 - 1) * q^2 + (-b3 + b2) * q^3 + (b2 + b1) * q^4 + (b1 - 2) * q^6 + (b3 - b2 - 2) * q^7 + (b3 - b1 - 1) * q^8 + (2*b3 - b2 - b1 + 1) * q^11 + (2*b3 + b2 - b1 + 2) * q^12 + (b3 + 2*b2 + b1 - 2) * q^13 + (2*b3 - b1 + 4) * q^14 + (2*b3 - 2*b2 + 2) * q^16 + (-b3 - 2*b2 - b1 - 4) * q^17 - 6 * q^19 + (2*b3 - 2*b2 + 3) * q^21 + (-2*b2 - b1 + 2) * q^22 + (b3 + 2*b2 + b1 - 1) * q^23 + (b3 - 4*b2 - b1 - 1) * q^24 + (5*b3 - 2*b2 - 2*b1 + 1) * q^26 + (-3*b3 + 3*b2) * q^27 + (-2*b3 - 3*b2 - b1 - 2) * q^28 + (-2*b3 - 4*b2 - 2*b1 - 3) * q^29 - 6 * q^31 + (-2*b3 - 2*b1 + 2) * q^32 + (-b3 - 2*b2 - b1 + 2) * q^33 + (b3 + 2*b2 + 2*b1 + 5) * q^34 + (-b3 - 2*b2 + 3*b1 - 3) * q^37 + (6*b3 + 6) * q^38 + (6*b3 - 3*b2 - 3*b1 + 3) * q^39 + (-2*b3 + 2*b2) * q^41 + (-3*b3 - 2*b1 + 1) * q^42 + 2 * q^43 + (-4*b3 + b2 + b1) * q^44 + (4*b3 - 2*b2 - 2*b1) * q^46 + (-b3 + b2) * q^47 + (-2*b3 + 2*b2 + 6) * q^48 + (-4*b3 + 4*b2 + 1) * q^49 + (-3*b2 + 3*b1 - 3) * q^51 + (2*b3 - 5*b2 - 3*b1 + 6) * q^52 + (-b3 + b1 - 1) * q^53 + (3*b1 - 6) * q^54 + (-3*b3 + 4*b2 + 3*b1 + 3) * q^56 + (6*b3 - 6*b2) * q^57 + (-3*b3 + 4*b2 + 4*b1 + 5) * q^58 + (b3 + 2*b2 + b1 + 1) * q^59 + (-b3 - 2*b2 + 3*b1 - 3) * q^61 + (6*b3 + 6) * q^62 + (-4*b3 + 4*b1 - 4) * q^64 + (-5*b3 + 2*b2 + 2*b1 - 1) * q^66 + (b3 + 2*b2 + b1 + 1) * q^67 + (-2*b3 - b2 - 3*b1 - 6) * q^68 + (5*b3 - 2*b2 - 3*b1 + 3) * q^69 + (2*b2 - 2*b1 + 2) * q^71 + (-2*b3 - 4*b2 - 2*b1 - 8) * q^73 + (6*b2 - 2*b1 + 4) * q^74 + (-6*b2 - 6*b1) * q^76 + (-3*b3 + 4*b2 + 3*b1 - 4) * q^77 + (-6*b2 - 3*b1 + 6) * q^78 + (2*b3 - 5*b2 + 3*b1 - 3) * q^79 - 9 * q^81 + (2*b1 - 4) * q^82 + (-4*b3 - 2*b2 + 6*b1 - 6) * q^83 + (-4*b3 + b2 + 5*b1 - 4) * q^84 + (-2*b3 - 2) * q^86 + (-5*b3 - b2 + 6*b1 - 6) * q^87 + (-3*b3 + 4*b2 + 3*b1 - 5) * q^88 + (5*b3 - 2*b2 - 3*b1 + 3) * q^89 + (-8*b3 - b2 + b1 + 1) * q^91 + (2*b3 - 4*b2 - 2*b1 + 6) * q^92 + (6*b3 - 6*b2) * q^93 + (b1 - 2) * q^94 + (-6*b3 + 2*b1 - 10) * q^96 + (-3*b3 - 6*b2 - 3*b1 - 6) * q^97 + (-b3 + 4*b1 - 9) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{2} - 6 q^{6} - 8 q^{7} - 8 q^{8}+O(q^{10})$$ 4 * q - 2 * q^2 - 6 * q^6 - 8 * q^7 - 8 * q^8 $$4 q - 2 q^{2} - 6 q^{6} - 8 q^{7} - 8 q^{8} - 12 q^{13} + 10 q^{14} + 8 q^{16} - 12 q^{17} - 24 q^{19} + 12 q^{21} + 10 q^{22} - 8 q^{23} - 6 q^{26} - 4 q^{29} - 24 q^{31} + 8 q^{32} + 12 q^{33} + 18 q^{34} + 12 q^{38} + 6 q^{42} + 8 q^{43} + 8 q^{44} - 8 q^{46} + 24 q^{48} + 4 q^{49} + 24 q^{52} - 18 q^{54} + 16 q^{56} + 26 q^{58} + 12 q^{62} + 6 q^{66} - 24 q^{68} - 24 q^{73} - 12 q^{77} + 30 q^{78} - 36 q^{81} - 12 q^{82} - 4 q^{86} - 16 q^{88} + 24 q^{91} + 24 q^{92} - 6 q^{94} - 24 q^{96} - 12 q^{97} - 26 q^{98}+O(q^{100})$$ 4 * q - 2 * q^2 - 6 * q^6 - 8 * q^7 - 8 * q^8 - 12 * q^13 + 10 * q^14 + 8 * q^16 - 12 * q^17 - 24 * q^19 + 12 * q^21 + 10 * q^22 - 8 * q^23 - 6 * q^26 - 4 * q^29 - 24 * q^31 + 8 * q^32 + 12 * q^33 + 18 * q^34 + 12 * q^38 + 6 * q^42 + 8 * q^43 + 8 * q^44 - 8 * q^46 + 24 * q^48 + 4 * q^49 + 24 * q^52 - 18 * q^54 + 16 * q^56 + 26 * q^58 + 12 * q^62 + 6 * q^66 - 24 * q^68 - 24 * q^73 - 12 * q^77 + 30 * q^78 - 36 * q^81 - 12 * q^82 - 4 * q^86 - 16 * q^88 + 24 * q^91 + 24 * q^92 - 6 * q^94 - 24 * q^96 - 12 * q^97 - 26 * q^98

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12}$$ v^3 + v^2 + v $$\beta_{2}$$ $$=$$ $$-\zeta_{12}^{3} - \zeta_{12}^{2} + \zeta_{12}$$ -v^3 - v^2 + v $$\beta_{3}$$ $$=$$ $$-\zeta_{12}^{3} + \zeta_{12}^{2} + \zeta_{12} - 1$$ -v^3 + v^2 + v - 1
 $$\zeta_{12}$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$( \beta_{3} - \beta_{2} + 1 ) / 2$$ (b3 - b2 + 1) / 2 $$\zeta_{12}^{3}$$ $$=$$ $$( -\beta_{3} + \beta _1 - 1 ) / 2$$ (-b3 + b1 - 1) / 2

## 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.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
699.1
 0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i −0.866025 + 0.500000i
−1.36603 0.366025i 1.73205i 1.73205 + 1.00000i 0 −0.633975 + 2.36603i −2.00000 + 1.73205i −2.00000 2.00000i 0 0
699.2 −1.36603 + 0.366025i 1.73205i 1.73205 1.00000i 0 −0.633975 2.36603i −2.00000 1.73205i −2.00000 + 2.00000i 0 0
699.3 0.366025 1.36603i 1.73205i −1.73205 1.00000i 0 −2.36603 0.633975i −2.00000 + 1.73205i −2.00000 + 2.00000i 0 0
699.4 0.366025 + 1.36603i 1.73205i −1.73205 + 1.00000i 0 −2.36603 + 0.633975i −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
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.b 4
4.b odd 2 1 700.2.c.f 4
5.b even 2 1 700.2.c.e 4
5.c odd 4 1 140.2.g.a 4
5.c odd 4 1 700.2.g.f 4
7.b odd 2 1 700.2.c.c 4
15.e even 4 1 1260.2.c.a 4
20.d odd 2 1 700.2.c.c 4
20.e even 4 1 140.2.g.b yes 4
20.e even 4 1 700.2.g.g 4
28.d even 2 1 700.2.c.e 4
35.c odd 2 1 700.2.c.f 4
35.f even 4 1 140.2.g.b yes 4
35.f even 4 1 700.2.g.g 4
35.k even 12 1 980.2.o.b 4
35.k even 12 1 980.2.o.d 4
35.l odd 12 1 980.2.o.a 4
35.l odd 12 1 980.2.o.c 4
40.i odd 4 1 2240.2.k.a 4
40.k even 4 1 2240.2.k.b 4
60.l odd 4 1 1260.2.c.b 4
105.k odd 4 1 1260.2.c.b 4
140.c even 2 1 inner 700.2.c.b 4
140.j odd 4 1 140.2.g.a 4
140.j odd 4 1 700.2.g.f 4
140.w even 12 1 980.2.o.b 4
140.w even 12 1 980.2.o.d 4
140.x odd 12 1 980.2.o.a 4
140.x odd 12 1 980.2.o.c 4
280.s even 4 1 2240.2.k.b 4
280.y odd 4 1 2240.2.k.a 4
420.w even 4 1 1260.2.c.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.a 4 5.c odd 4 1
140.2.g.a 4 140.j odd 4 1
140.2.g.b yes 4 20.e even 4 1
140.2.g.b yes 4 35.f even 4 1
700.2.c.b 4 1.a even 1 1 trivial
700.2.c.b 4 140.c even 2 1 inner
700.2.c.c 4 7.b odd 2 1
700.2.c.c 4 20.d odd 2 1
700.2.c.e 4 5.b even 2 1
700.2.c.e 4 28.d even 2 1
700.2.c.f 4 4.b odd 2 1
700.2.c.f 4 35.c odd 2 1
700.2.g.f 4 5.c odd 4 1
700.2.g.f 4 140.j odd 4 1
700.2.g.g 4 20.e even 4 1
700.2.g.g 4 35.f even 4 1
980.2.o.a 4 35.l odd 12 1
980.2.o.a 4 140.x odd 12 1
980.2.o.b 4 35.k even 12 1
980.2.o.b 4 140.w even 12 1
980.2.o.c 4 35.l odd 12 1
980.2.o.c 4 140.x odd 12 1
980.2.o.d 4 35.k even 12 1
980.2.o.d 4 140.w even 12 1
1260.2.c.a 4 15.e even 4 1
1260.2.c.a 4 420.w even 4 1
1260.2.c.b 4 60.l odd 4 1
1260.2.c.b 4 105.k odd 4 1
2240.2.k.a 4 40.i odd 4 1
2240.2.k.a 4 280.y odd 4 1
2240.2.k.b 4 40.k even 4 1
2240.2.k.b 4 280.s even 4 1

## Hecke kernels

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

 $$T_{3}^{2} + 3$$ T3^2 + 3 $$T_{11}^{4} + 14T_{11}^{2} + 1$$ T11^4 + 14*T11^2 + 1 $$T_{13}^{2} + 6T_{13} - 3$$ T13^2 + 6*T13 - 3 $$T_{19} + 6$$ T19 + 6 $$T_{23}^{2} + 4T_{23} - 8$$ T23^2 + 4*T23 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 2 T^{3} + 2 T^{2} + 4 T + 4$$
$3$ $$(T^{2} + 3)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 4 T + 7)^{2}$$
$11$ $$T^{4} + 14T^{2} + 1$$
$13$ $$(T^{2} + 6 T - 3)^{2}$$
$17$ $$(T^{2} + 6 T - 3)^{2}$$
$19$ $$(T + 6)^{4}$$
$23$ $$(T^{2} + 4 T - 8)^{2}$$
$29$ $$(T^{2} + 2 T - 47)^{2}$$
$31$ $$(T + 6)^{4}$$
$37$ $$T^{4} + 96T^{2} + 576$$
$41$ $$(T^{2} + 12)^{2}$$
$43$ $$(T - 2)^{4}$$
$47$ $$(T^{2} + 3)^{2}$$
$53$ $$(T^{2} + 4)^{2}$$
$59$ $$(T^{2} - 12)^{2}$$
$61$ $$T^{4} + 96T^{2} + 576$$
$67$ $$(T^{2} - 12)^{2}$$
$71$ $$T^{4} + 56T^{2} + 16$$
$73$ $$(T^{2} + 12 T - 12)^{2}$$
$79$ $$T^{4} + 222T^{2} + 1521$$
$83$ $$T^{4} + 312 T^{2} + 17424$$
$89$ $$T^{4} + 96T^{2} + 576$$
$97$ $$(T^{2} + 6 T - 99)^{2}$$