# Properties

 Label 700.2.g.f Level $700$ Weight $2$ Character orbit 700.g Analytic conductor $5.590$ Analytic rank $0$ 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(251,700)]

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, 0, 1]))

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

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

chi := DirichletCharacter("700.251");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 700.g (of order $$2$$, degree $$1$$, 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: $$0$$ 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: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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.

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

Basis of coefficient ring

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1
 −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i 0.866025 + 0.500000i
−0.366025 1.36603i 1.73205 −1.73205 + 1.00000i 0 −0.633975 2.36603i 1.73205 2.00000i 2.00000 + 2.00000i 0 0
251.2 −0.366025 + 1.36603i 1.73205 −1.73205 1.00000i 0 −0.633975 + 2.36603i 1.73205 + 2.00000i 2.00000 2.00000i 0 0
251.3 1.36603 0.366025i −1.73205 1.73205 1.00000i 0 −2.36603 + 0.633975i −1.73205 + 2.00000i 2.00000 2.00000i 0 0
251.4 1.36603 + 0.366025i −1.73205 1.73205 + 1.00000i 0 −2.36603 0.633975i −1.73205 2.00000i 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
28.d 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.g.f 4
4.b odd 2 1 700.2.g.g 4
5.b even 2 1 140.2.g.a 4
5.c odd 4 1 700.2.c.b 4
5.c odd 4 1 700.2.c.e 4
7.b odd 2 1 700.2.g.g 4
15.d odd 2 1 1260.2.c.a 4
20.d odd 2 1 140.2.g.b yes 4
20.e even 4 1 700.2.c.c 4
20.e even 4 1 700.2.c.f 4
28.d even 2 1 inner 700.2.g.f 4
35.c odd 2 1 140.2.g.b yes 4
35.f even 4 1 700.2.c.c 4
35.f even 4 1 700.2.c.f 4
35.i odd 6 1 980.2.o.b 4
35.i odd 6 1 980.2.o.d 4
35.j even 6 1 980.2.o.a 4
35.j even 6 1 980.2.o.c 4
40.e odd 2 1 2240.2.k.b 4
40.f even 2 1 2240.2.k.a 4
60.h even 2 1 1260.2.c.b 4
105.g even 2 1 1260.2.c.b 4
140.c even 2 1 140.2.g.a 4
140.j odd 4 1 700.2.c.b 4
140.j odd 4 1 700.2.c.e 4
140.p odd 6 1 980.2.o.b 4
140.p odd 6 1 980.2.o.d 4
140.s even 6 1 980.2.o.a 4
140.s even 6 1 980.2.o.c 4
280.c odd 2 1 2240.2.k.b 4
280.n even 2 1 2240.2.k.a 4
420.o odd 2 1 1260.2.c.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
140.2.g.a 4 5.b even 2 1
140.2.g.a 4 140.c even 2 1
140.2.g.b yes 4 20.d odd 2 1
140.2.g.b yes 4 35.c odd 2 1
700.2.c.b 4 5.c odd 4 1
700.2.c.b 4 140.j odd 4 1
700.2.c.c 4 20.e even 4 1
700.2.c.c 4 35.f even 4 1
700.2.c.e 4 5.c odd 4 1
700.2.c.e 4 140.j odd 4 1
700.2.c.f 4 20.e even 4 1
700.2.c.f 4 35.f even 4 1
700.2.g.f 4 1.a even 1 1 trivial
700.2.g.f 4 28.d even 2 1 inner
700.2.g.g 4 4.b odd 2 1
700.2.g.g 4 7.b odd 2 1
980.2.o.a 4 35.j even 6 1
980.2.o.a 4 140.s even 6 1
980.2.o.b 4 35.i odd 6 1
980.2.o.b 4 140.p odd 6 1
980.2.o.c 4 35.j even 6 1
980.2.o.c 4 140.s even 6 1
980.2.o.d 4 35.i odd 6 1
980.2.o.d 4 140.p odd 6 1
1260.2.c.a 4 15.d odd 2 1
1260.2.c.a 4 420.o odd 2 1
1260.2.c.b 4 60.h even 2 1
1260.2.c.b 4 105.g even 2 1
2240.2.k.a 4 40.f even 2 1
2240.2.k.a 4 280.n even 2 1
2240.2.k.b 4 40.e odd 2 1
2240.2.k.b 4 280.c odd 2 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 $$T_{19} - 6$$ T19 - 6 $$T_{37}^{2} - 12T_{37} + 24$$ T37^2 - 12*T37 + 24

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} - 2 T^{3} + 2 T^{2} - 4 T + 4$$
$3$ $$(T^{2} - 3)^{2}$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 2T^{2} + 49$$
$11$ $$T^{4} + 14T^{2} + 1$$
$13$ $$T^{4} + 42T^{2} + 9$$
$17$ $$T^{4} + 42T^{2} + 9$$
$19$ $$(T - 6)^{4}$$
$23$ $$T^{4} + 32T^{2} + 64$$
$29$ $$(T^{2} - 2 T - 47)^{2}$$
$31$ $$(T + 6)^{4}$$
$37$ $$(T^{2} - 12 T + 24)^{2}$$
$41$ $$(T^{2} + 12)^{2}$$
$43$ $$(T^{2} + 4)^{2}$$
$47$ $$(T^{2} - 3)^{2}$$
$53$ $$(T + 2)^{4}$$
$59$ $$(T^{2} - 12)^{2}$$
$61$ $$T^{4} + 96T^{2} + 576$$
$67$ $$(T^{2} + 12)^{2}$$
$71$ $$T^{4} + 56T^{2} + 16$$
$73$ $$T^{4} + 168T^{2} + 144$$
$79$ $$T^{4} + 222T^{2} + 1521$$
$83$ $$(T^{2} + 24 T + 132)^{2}$$
$89$ $$T^{4} + 96T^{2} + 576$$
$97$ $$T^{4} + 234T^{2} + 9801$$