# Properties

 Label 275.2.b.c Level $275$ Weight $2$ Character orbit 275.b Analytic conductor $2.196$ 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] = [275,2,Mod(199,275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(275, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

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

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

chi := DirichletCharacter("275.199");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$275 = 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 275.b (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: $$2.19588605559$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{13})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 7x^{2} + 9$$ x^4 + 7*x^2 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{3} + (\beta_{3} - 2) q^{4} - 3 q^{6} + ( - 2 \beta_{2} + \beta_1) q^{7} + 3 \beta_{2} q^{8} - \beta_{3} q^{9}+O(q^{10})$$ q + b1 * q^2 + (b2 + b1) * q^3 + (b3 - 2) * q^4 - 3 * q^6 + (-2*b2 + b1) * q^7 + 3*b2 * q^8 - b3 * q^9 $$q + \beta_1 q^{2} + (\beta_{2} + \beta_1) q^{3} + (\beta_{3} - 2) q^{4} - 3 q^{6} + ( - 2 \beta_{2} + \beta_1) q^{7} + 3 \beta_{2} q^{8} - \beta_{3} q^{9} - q^{11} + (2 \beta_{2} - \beta_1) q^{12} + 5 \beta_{2} q^{13} + (3 \beta_{3} - 6) q^{14} + ( - \beta_{3} - 1) q^{16} + ( - 3 \beta_{2} - 3 \beta_1) q^{17} - 3 \beta_{2} q^{18} + q^{19} + (2 \beta_{3} - 3) q^{21} - \beta_1 q^{22} + ( - 6 \beta_{2} - \beta_1) q^{23} - 3 \beta_{3} q^{24} + ( - 5 \beta_{3} + 5) q^{26} + ( - \beta_{2} + 2 \beta_1) q^{27} + (5 \beta_{2} - 4 \beta_1) q^{28} + ( - 3 \beta_{3} + 6) q^{29} + (4 \beta_{3} + 1) q^{31} + (3 \beta_{2} - \beta_1) q^{32} + ( - \beta_{2} - \beta_1) q^{33} + 9 q^{34} + (\beta_{3} - 3) q^{36} + ( - 5 \beta_{2} + 2 \beta_1) q^{37} + \beta_1 q^{38} - 5 \beta_{3} q^{39} + ( - 2 \beta_{3} - 1) q^{41} + (6 \beta_{2} - 3 \beta_1) q^{42} + (2 \beta_{2} + 4 \beta_1) q^{43} + ( - \beta_{3} + 2) q^{44} + (5 \beta_{3} - 2) q^{46} + 3 \beta_{2} q^{47} + ( - 5 \beta_{2} - 2 \beta_1) q^{48} + (5 \beta_{3} - 5) q^{49} + (3 \beta_{3} + 9) q^{51} + ( - 5 \beta_{2} + 5 \beta_1) q^{52} - \beta_1 q^{53} + (3 \beta_{3} - 9) q^{54} + ( - 3 \beta_{3} + 9) q^{56} + (\beta_{2} + \beta_1) q^{57} + ( - 9 \beta_{2} + 6 \beta_1) q^{58} + (4 \beta_{3} + 5) q^{59} + ( - 3 \beta_{3} - 1) q^{61} + (12 \beta_{2} + \beta_1) q^{62} + ( - \beta_{2} + 2 \beta_1) q^{63} + ( - 6 \beta_{3} + 5) q^{64} + 3 q^{66} + 4 \beta_{2} q^{67} + ( - 6 \beta_{2} + 3 \beta_1) q^{68} + (6 \beta_{3} + 3) q^{69} + ( - 2 \beta_{3} + 2) q^{71} + ( - 3 \beta_{2} - 3 \beta_1) q^{72} + ( - 4 \beta_{2} - 3 \beta_1) q^{73} + (7 \beta_{3} - 13) q^{74} + (\beta_{3} - 2) q^{76} + (2 \beta_{2} - \beta_1) q^{77} - 15 \beta_{2} q^{78} + (3 \beta_{3} + 4) q^{79} + ( - 2 \beta_{3} - 6) q^{81} + ( - 6 \beta_{2} - \beta_1) q^{82} + (3 \beta_{2} - 5 \beta_1) q^{83} + ( - 5 \beta_{3} + 12) q^{84} + (2 \beta_{3} - 14) q^{86} + ( - 6 \beta_{2} + 3 \beta_1) q^{87} - 3 \beta_{2} q^{88} + (\beta_{3} - 4) q^{89} + ( - 5 \beta_{3} + 15) q^{91} + (3 \beta_{2} - 4 \beta_1) q^{92} + (17 \beta_{2} + 5 \beta_1) q^{93} + ( - 3 \beta_{3} + 3) q^{94} + ( - 3 \beta_{3} + 3) q^{96} + ( - 14 \beta_{2} - \beta_1) q^{97} + (15 \beta_{2} - 5 \beta_1) q^{98} + \beta_{3} q^{99}+O(q^{100})$$ q + b1 * q^2 + (b2 + b1) * q^3 + (b3 - 2) * q^4 - 3 * q^6 + (-2*b2 + b1) * q^7 + 3*b2 * q^8 - b3 * q^9 - q^11 + (2*b2 - b1) * q^12 + 5*b2 * q^13 + (3*b3 - 6) * q^14 + (-b3 - 1) * q^16 + (-3*b2 - 3*b1) * q^17 - 3*b2 * q^18 + q^19 + (2*b3 - 3) * q^21 - b1 * q^22 + (-6*b2 - b1) * q^23 - 3*b3 * q^24 + (-5*b3 + 5) * q^26 + (-b2 + 2*b1) * q^27 + (5*b2 - 4*b1) * q^28 + (-3*b3 + 6) * q^29 + (4*b3 + 1) * q^31 + (3*b2 - b1) * q^32 + (-b2 - b1) * q^33 + 9 * q^34 + (b3 - 3) * q^36 + (-5*b2 + 2*b1) * q^37 + b1 * q^38 - 5*b3 * q^39 + (-2*b3 - 1) * q^41 + (6*b2 - 3*b1) * q^42 + (2*b2 + 4*b1) * q^43 + (-b3 + 2) * q^44 + (5*b3 - 2) * q^46 + 3*b2 * q^47 + (-5*b2 - 2*b1) * q^48 + (5*b3 - 5) * q^49 + (3*b3 + 9) * q^51 + (-5*b2 + 5*b1) * q^52 - b1 * q^53 + (3*b3 - 9) * q^54 + (-3*b3 + 9) * q^56 + (b2 + b1) * q^57 + (-9*b2 + 6*b1) * q^58 + (4*b3 + 5) * q^59 + (-3*b3 - 1) * q^61 + (12*b2 + b1) * q^62 + (-b2 + 2*b1) * q^63 + (-6*b3 + 5) * q^64 + 3 * q^66 + 4*b2 * q^67 + (-6*b2 + 3*b1) * q^68 + (6*b3 + 3) * q^69 + (-2*b3 + 2) * q^71 + (-3*b2 - 3*b1) * q^72 + (-4*b2 - 3*b1) * q^73 + (7*b3 - 13) * q^74 + (b3 - 2) * q^76 + (2*b2 - b1) * q^77 - 15*b2 * q^78 + (3*b3 + 4) * q^79 + (-2*b3 - 6) * q^81 + (-6*b2 - b1) * q^82 + (3*b2 - 5*b1) * q^83 + (-5*b3 + 12) * q^84 + (2*b3 - 14) * q^86 + (-6*b2 + 3*b1) * q^87 - 3*b2 * q^88 + (b3 - 4) * q^89 + (-5*b3 + 15) * q^91 + (3*b2 - 4*b1) * q^92 + (17*b2 + 5*b1) * q^93 + (-3*b3 + 3) * q^94 + (-3*b3 + 3) * q^96 + (-14*b2 - b1) * q^97 + (15*b2 - 5*b1) * q^98 + b3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 6 q^{4} - 12 q^{6} - 2 q^{9}+O(q^{10})$$ 4 * q - 6 * q^4 - 12 * q^6 - 2 * q^9 $$4 q - 6 q^{4} - 12 q^{6} - 2 q^{9} - 4 q^{11} - 18 q^{14} - 6 q^{16} + 4 q^{19} - 8 q^{21} - 6 q^{24} + 10 q^{26} + 18 q^{29} + 12 q^{31} + 36 q^{34} - 10 q^{36} - 10 q^{39} - 8 q^{41} + 6 q^{44} + 2 q^{46} - 10 q^{49} + 42 q^{51} - 30 q^{54} + 30 q^{56} + 28 q^{59} - 10 q^{61} + 8 q^{64} + 12 q^{66} + 24 q^{69} + 4 q^{71} - 38 q^{74} - 6 q^{76} + 22 q^{79} - 28 q^{81} + 38 q^{84} - 52 q^{86} - 14 q^{89} + 50 q^{91} + 6 q^{94} + 6 q^{96} + 2 q^{99}+O(q^{100})$$ 4 * q - 6 * q^4 - 12 * q^6 - 2 * q^9 - 4 * q^11 - 18 * q^14 - 6 * q^16 + 4 * q^19 - 8 * q^21 - 6 * q^24 + 10 * q^26 + 18 * q^29 + 12 * q^31 + 36 * q^34 - 10 * q^36 - 10 * q^39 - 8 * q^41 + 6 * q^44 + 2 * q^46 - 10 * q^49 + 42 * q^51 - 30 * q^54 + 30 * q^56 + 28 * q^59 - 10 * q^61 + 8 * q^64 + 12 * q^66 + 24 * q^69 + 4 * q^71 - 38 * q^74 - 6 * q^76 + 22 * q^79 - 28 * q^81 + 38 * q^84 - 52 * q^86 - 14 * q^89 + 50 * q^91 + 6 * q^94 + 6 * q^96 + 2 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 7x^{2} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 4\nu ) / 3$$ (v^3 + 4*v) / 3 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 4$$ v^2 + 4
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 4$$ b3 - 4 $$\nu^{3}$$ $$=$$ $$3\beta_{2} - 4\beta_1$$ 3*b2 - 4*b1

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/275\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$177$$ $$\chi(n)$$ $$1$$ $$-1$$

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
199.1
 − 2.30278i − 1.30278i 1.30278i 2.30278i
2.30278i 1.30278i −3.30278 0 −3.00000 4.30278i 3.00000i 1.30278 0
199.2 1.30278i 2.30278i 0.302776 0 −3.00000 0.697224i 3.00000i −2.30278 0
199.3 1.30278i 2.30278i 0.302776 0 −3.00000 0.697224i 3.00000i −2.30278 0
199.4 2.30278i 1.30278i −3.30278 0 −3.00000 4.30278i 3.00000i 1.30278 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 275.2.b.c 4
3.b odd 2 1 2475.2.c.k 4
4.b odd 2 1 4400.2.b.y 4
5.b even 2 1 inner 275.2.b.c 4
5.c odd 4 1 275.2.a.e 2
5.c odd 4 1 275.2.a.f yes 2
15.d odd 2 1 2475.2.c.k 4
15.e even 4 1 2475.2.a.o 2
15.e even 4 1 2475.2.a.t 2
20.d odd 2 1 4400.2.b.y 4
20.e even 4 1 4400.2.a.bh 2
20.e even 4 1 4400.2.a.bs 2
55.e even 4 1 3025.2.a.h 2
55.e even 4 1 3025.2.a.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.e 2 5.c odd 4 1
275.2.a.f yes 2 5.c odd 4 1
275.2.b.c 4 1.a even 1 1 trivial
275.2.b.c 4 5.b even 2 1 inner
2475.2.a.o 2 15.e even 4 1
2475.2.a.t 2 15.e even 4 1
2475.2.c.k 4 3.b odd 2 1
2475.2.c.k 4 15.d odd 2 1
3025.2.a.h 2 55.e even 4 1
3025.2.a.n 2 55.e even 4 1
4400.2.a.bh 2 20.e even 4 1
4400.2.a.bs 2 20.e even 4 1
4400.2.b.y 4 4.b odd 2 1
4400.2.b.y 4 20.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 7T_{2}^{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(275, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 7T^{2} + 9$$
$3$ $$T^{4} + 7T^{2} + 9$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 19T^{2} + 9$$
$11$ $$(T + 1)^{4}$$
$13$ $$(T^{2} + 25)^{2}$$
$17$ $$T^{4} + 63T^{2} + 729$$
$19$ $$(T - 1)^{4}$$
$23$ $$T^{4} + 67T^{2} + 729$$
$29$ $$(T^{2} - 9 T - 9)^{2}$$
$31$ $$(T^{2} - 6 T - 43)^{2}$$
$37$ $$T^{4} + 98T^{2} + 529$$
$41$ $$(T^{2} + 4 T - 9)^{2}$$
$43$ $$(T^{2} + 52)^{2}$$
$47$ $$(T^{2} + 9)^{2}$$
$53$ $$T^{4} + 7T^{2} + 9$$
$59$ $$(T^{2} - 14 T - 3)^{2}$$
$61$ $$(T^{2} + 5 T - 23)^{2}$$
$67$ $$(T^{2} + 16)^{2}$$
$71$ $$(T^{2} - 2 T - 12)^{2}$$
$73$ $$T^{4} + 71T^{2} + 529$$
$79$ $$(T^{2} - 11 T + 1)^{2}$$
$83$ $$T^{4} + 223T^{2} + 2601$$
$89$ $$(T^{2} + 7 T + 9)^{2}$$
$97$ $$T^{4} + 371 T^{2} + 32041$$