# Properties

 Label 275.2.b.d 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: $$\Q(\zeta_{8})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{4} + 1$$ x^4 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$2^{3}$$ Twist minimal: no (minimal twist has level 55) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{2} + ( - \beta_{2} + \beta_1) q^{3} + ( - \beta_{3} - 1) q^{4} + ( - \beta_{3} - 4) q^{6} + ( - \beta_{2} - \beta_1) q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8} - 5 q^{9}+O(q^{10})$$ q + b1 * q^2 + (-b2 + b1) * q^3 + (-b3 - 1) * q^4 + (-b3 - 4) * q^6 + (-b2 - b1) * q^7 + (-b2 - 2*b1) * q^8 - 5 * q^9 $$q + \beta_1 q^{2} + ( - \beta_{2} + \beta_1) q^{3} + ( - \beta_{3} - 1) q^{4} + ( - \beta_{3} - 4) q^{6} + ( - \beta_{2} - \beta_1) q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8} - 5 q^{9} + q^{11} + ( - 3 \beta_{2} - 5 \beta_1) q^{12} + (3 \beta_{2} + \beta_1) q^{13} + (\beta_{3} + 2) q^{14} + 3 q^{16} + (\beta_{2} + 3 \beta_1) q^{17} - 5 \beta_1 q^{18} + 2 \beta_{3} q^{21} + \beta_1 q^{22} + ( - \beta_{2} + \beta_1) q^{23} + (3 \beta_{3} + 4) q^{24} - \beta_{3} q^{26} + (2 \beta_{2} - 2 \beta_1) q^{27} + ( - \beta_{2} + 3 \beta_1) q^{28} + (2 \beta_{3} - 2) q^{29} + ( - 2 \beta_{2} - \beta_1) q^{32} + ( - \beta_{2} + \beta_1) q^{33} + ( - 3 \beta_{3} - 8) q^{34} + (5 \beta_{3} + 5) q^{36} + (\beta_{2} - 3 \beta_1) q^{37} + ( - 4 \beta_{3} + 8) q^{39} + 6 q^{41} + (2 \beta_{2} + 6 \beta_1) q^{42} + (3 \beta_{2} + 3 \beta_1) q^{43} + ( - \beta_{3} - 1) q^{44} + ( - \beta_{3} - 4) q^{46} + ( - \beta_{2} + \beta_1) q^{47} + ( - 3 \beta_{2} + 3 \beta_1) q^{48} + 3 q^{49} + ( - 4 \beta_{3} - 8) q^{51} + (5 \beta_{2} - \beta_1) q^{52} + ( - \beta_{2} - 5 \beta_1) q^{53} + (2 \beta_{3} + 8) q^{54} + ( - \beta_{3} - 6) q^{56} + (2 \beta_{2} + 4 \beta_1) q^{58} + ( - 2 \beta_{3} + 4) q^{59} + ( - 4 \beta_{3} + 2) q^{61} + (5 \beta_{2} + 5 \beta_1) q^{63} + (\beta_{3} + 7) q^{64} + ( - \beta_{3} - 4) q^{66} + ( - \beta_{2} + 5 \beta_1) q^{67} + ( - \beta_{2} - 11 \beta_1) q^{68} - 8 q^{69} + 4 \beta_{3} q^{71} + (5 \beta_{2} + 10 \beta_1) q^{72} + (3 \beta_{2} + \beta_1) q^{73} + (3 \beta_{3} + 10) q^{74} + ( - \beta_{2} - \beta_1) q^{77} + ( - 4 \beta_{2} - 4 \beta_1) q^{78} - 4 q^{79} + q^{81} + 6 \beta_1 q^{82} + (3 \beta_{2} + 3 \beta_1) q^{83} + ( - 2 \beta_{3} - 16) q^{84} + ( - 3 \beta_{3} - 6) q^{86} + (10 \beta_{2} + 6 \beta_1) q^{87} + ( - \beta_{2} - 2 \beta_1) q^{88} + (4 \beta_{3} + 2) q^{89} + ( - 2 \beta_{3} + 8) q^{91} + ( - 3 \beta_{2} - 5 \beta_1) q^{92} + ( - \beta_{3} - 4) q^{94} + (3 \beta_{3} - 4) q^{96} + ( - 3 \beta_{2} + \beta_1) q^{97} + 3 \beta_1 q^{98} - 5 q^{99}+O(q^{100})$$ q + b1 * q^2 + (-b2 + b1) * q^3 + (-b3 - 1) * q^4 + (-b3 - 4) * q^6 + (-b2 - b1) * q^7 + (-b2 - 2*b1) * q^8 - 5 * q^9 + q^11 + (-3*b2 - 5*b1) * q^12 + (3*b2 + b1) * q^13 + (b3 + 2) * q^14 + 3 * q^16 + (b2 + 3*b1) * q^17 - 5*b1 * q^18 + 2*b3 * q^21 + b1 * q^22 + (-b2 + b1) * q^23 + (3*b3 + 4) * q^24 - b3 * q^26 + (2*b2 - 2*b1) * q^27 + (-b2 + 3*b1) * q^28 + (2*b3 - 2) * q^29 + (-2*b2 - b1) * q^32 + (-b2 + b1) * q^33 + (-3*b3 - 8) * q^34 + (5*b3 + 5) * q^36 + (b2 - 3*b1) * q^37 + (-4*b3 + 8) * q^39 + 6 * q^41 + (2*b2 + 6*b1) * q^42 + (3*b2 + 3*b1) * q^43 + (-b3 - 1) * q^44 + (-b3 - 4) * q^46 + (-b2 + b1) * q^47 + (-3*b2 + 3*b1) * q^48 + 3 * q^49 + (-4*b3 - 8) * q^51 + (5*b2 - b1) * q^52 + (-b2 - 5*b1) * q^53 + (2*b3 + 8) * q^54 + (-b3 - 6) * q^56 + (2*b2 + 4*b1) * q^58 + (-2*b3 + 4) * q^59 + (-4*b3 + 2) * q^61 + (5*b2 + 5*b1) * q^63 + (b3 + 7) * q^64 + (-b3 - 4) * q^66 + (-b2 + 5*b1) * q^67 + (-b2 - 11*b1) * q^68 - 8 * q^69 + 4*b3 * q^71 + (5*b2 + 10*b1) * q^72 + (3*b2 + b1) * q^73 + (3*b3 + 10) * q^74 + (-b2 - b1) * q^77 + (-4*b2 - 4*b1) * q^78 - 4 * q^79 + q^81 + 6*b1 * q^82 + (3*b2 + 3*b1) * q^83 + (-2*b3 - 16) * q^84 + (-3*b3 - 6) * q^86 + (10*b2 + 6*b1) * q^87 + (-b2 - 2*b1) * q^88 + (4*b3 + 2) * q^89 + (-2*b3 + 8) * q^91 + (-3*b2 - 5*b1) * q^92 + (-b3 - 4) * q^94 + (3*b3 - 4) * q^96 + (-3*b2 + b1) * q^97 + 3*b1 * q^98 - 5 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 4 q^{4} - 16 q^{6} - 20 q^{9}+O(q^{10})$$ 4 * q - 4 * q^4 - 16 * q^6 - 20 * q^9 $$4 q - 4 q^{4} - 16 q^{6} - 20 q^{9} + 4 q^{11} + 8 q^{14} + 12 q^{16} + 16 q^{24} - 8 q^{29} - 32 q^{34} + 20 q^{36} + 32 q^{39} + 24 q^{41} - 4 q^{44} - 16 q^{46} + 12 q^{49} - 32 q^{51} + 32 q^{54} - 24 q^{56} + 16 q^{59} + 8 q^{61} + 28 q^{64} - 16 q^{66} - 32 q^{69} + 40 q^{74} - 16 q^{79} + 4 q^{81} - 64 q^{84} - 24 q^{86} + 8 q^{89} + 32 q^{91} - 16 q^{94} - 16 q^{96} - 20 q^{99}+O(q^{100})$$ 4 * q - 4 * q^4 - 16 * q^6 - 20 * q^9 + 4 * q^11 + 8 * q^14 + 12 * q^16 + 16 * q^24 - 8 * q^29 - 32 * q^34 + 20 * q^36 + 32 * q^39 + 24 * q^41 - 4 * q^44 - 16 * q^46 + 12 * q^49 - 32 * q^51 + 32 * q^54 - 24 * q^56 + 16 * q^59 + 8 * q^61 + 28 * q^64 - 16 * q^66 - 32 * q^69 + 40 * q^74 - 16 * q^79 + 4 * q^81 - 64 * q^84 - 24 * q^86 + 8 * q^89 + 32 * q^91 - 16 * q^94 - 16 * q^96 - 20 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$\zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8}$$ v^3 + v^2 + v $$\beta_{2}$$ $$=$$ $$-\zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8}$$ -v^3 + v^2 - v $$\beta_{3}$$ $$=$$ $$-2\zeta_{8}^{3} + 2\zeta_{8}$$ -2*v^3 + 2*v
 $$\zeta_{8}$$ $$=$$ $$( \beta_{3} - \beta_{2} + \beta_1 ) / 4$$ (b3 - b2 + b1) / 4 $$\zeta_{8}^{2}$$ $$=$$ $$( \beta_{2} + \beta_1 ) / 2$$ (b2 + b1) / 2 $$\zeta_{8}^{3}$$ $$=$$ $$( -\beta_{3} - \beta_{2} + \beta_1 ) / 4$$ (-b3 - b2 + b1) / 4

## 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
 0.707107 − 0.707107i −0.707107 − 0.707107i −0.707107 + 0.707107i 0.707107 + 0.707107i
2.41421i 2.82843i −3.82843 0 −6.82843 2.00000i 4.41421i −5.00000 0
199.2 0.414214i 2.82843i 1.82843 0 −1.17157 2.00000i 1.58579i −5.00000 0
199.3 0.414214i 2.82843i 1.82843 0 −1.17157 2.00000i 1.58579i −5.00000 0
199.4 2.41421i 2.82843i −3.82843 0 −6.82843 2.00000i 4.41421i −5.00000 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.d 4
3.b odd 2 1 2475.2.c.l 4
4.b odd 2 1 4400.2.b.q 4
5.b even 2 1 inner 275.2.b.d 4
5.c odd 4 1 55.2.a.b 2
5.c odd 4 1 275.2.a.c 2
15.d odd 2 1 2475.2.c.l 4
15.e even 4 1 495.2.a.b 2
15.e even 4 1 2475.2.a.x 2
20.d odd 2 1 4400.2.b.q 4
20.e even 4 1 880.2.a.m 2
20.e even 4 1 4400.2.a.bn 2
35.f even 4 1 2695.2.a.f 2
40.i odd 4 1 3520.2.a.bn 2
40.k even 4 1 3520.2.a.bo 2
55.e even 4 1 605.2.a.d 2
55.e even 4 1 3025.2.a.o 2
55.k odd 20 4 605.2.g.f 8
55.l even 20 4 605.2.g.l 8
60.l odd 4 1 7920.2.a.ch 2
65.h odd 4 1 9295.2.a.g 2
165.l odd 4 1 5445.2.a.y 2
220.i odd 4 1 9680.2.a.bn 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b 2 5.c odd 4 1
275.2.a.c 2 5.c odd 4 1
275.2.b.d 4 1.a even 1 1 trivial
275.2.b.d 4 5.b even 2 1 inner
495.2.a.b 2 15.e even 4 1
605.2.a.d 2 55.e even 4 1
605.2.g.f 8 55.k odd 20 4
605.2.g.l 8 55.l even 20 4
880.2.a.m 2 20.e even 4 1
2475.2.a.x 2 15.e even 4 1
2475.2.c.l 4 3.b odd 2 1
2475.2.c.l 4 15.d odd 2 1
2695.2.a.f 2 35.f even 4 1
3025.2.a.o 2 55.e even 4 1
3520.2.a.bn 2 40.i odd 4 1
3520.2.a.bo 2 40.k even 4 1
4400.2.a.bn 2 20.e even 4 1
4400.2.b.q 4 4.b odd 2 1
4400.2.b.q 4 20.d odd 2 1
5445.2.a.y 2 165.l odd 4 1
7920.2.a.ch 2 60.l odd 4 1
9295.2.a.g 2 65.h odd 4 1
9680.2.a.bn 2 220.i odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 6T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(275, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4} + 6T^{2} + 1$$
$3$ $$(T^{2} + 8)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 4)^{2}$$
$11$ $$(T - 1)^{4}$$
$13$ $$T^{4} + 48T^{2} + 64$$
$17$ $$T^{4} + 48T^{2} + 64$$
$19$ $$T^{4}$$
$23$ $$(T^{2} + 8)^{2}$$
$29$ $$(T^{2} + 4 T - 28)^{2}$$
$31$ $$T^{4}$$
$37$ $$T^{4} + 72T^{2} + 784$$
$41$ $$(T - 6)^{4}$$
$43$ $$(T^{2} + 36)^{2}$$
$47$ $$(T^{2} + 8)^{2}$$
$53$ $$T^{4} + 136T^{2} + 16$$
$59$ $$(T^{2} - 8 T - 16)^{2}$$
$61$ $$(T^{2} - 4 T - 124)^{2}$$
$67$ $$T^{4} + 176T^{2} + 3136$$
$71$ $$(T^{2} - 128)^{2}$$
$73$ $$T^{4} + 48T^{2} + 64$$
$79$ $$(T + 4)^{4}$$
$83$ $$(T^{2} + 36)^{2}$$
$89$ $$(T^{2} - 4 T - 124)^{2}$$
$97$ $$T^{4} + 72T^{2} + 784$$