# Properties

 Label 605.2.a.d Level $605$ Weight $2$ Character orbit 605.a Self dual yes Analytic conductor $4.831$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [605,2,Mod(1,605)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(605, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("605.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.a (trivial)

## 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: yes Analytic conductor: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{2} - 2$$ x^2 - 2 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 55) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + (\beta - 1) q^{2} + 2 \beta q^{3} + ( - 2 \beta + 1) q^{4} - q^{5} + ( - 2 \beta + 4) q^{6} + 2 q^{7} + (\beta - 3) q^{8} + 5 q^{9}+O(q^{10})$$ q + (b - 1) * q^2 + 2*b * q^3 + (-2*b + 1) * q^4 - q^5 + (-2*b + 4) * q^6 + 2 * q^7 + (b - 3) * q^8 + 5 * q^9 $$q + (\beta - 1) q^{2} + 2 \beta q^{3} + ( - 2 \beta + 1) q^{4} - q^{5} + ( - 2 \beta + 4) q^{6} + 2 q^{7} + (\beta - 3) q^{8} + 5 q^{9} + ( - \beta + 1) q^{10} + (2 \beta - 8) q^{12} + (2 \beta + 4) q^{13} + (2 \beta - 2) q^{14} - 2 \beta q^{15} + 3 q^{16} + (2 \beta - 4) q^{17} + (5 \beta - 5) q^{18} + (2 \beta - 1) q^{20} + 4 \beta q^{21} + 2 \beta q^{23} + ( - 6 \beta + 4) q^{24} + q^{25} + 2 \beta q^{26} + 4 \beta q^{27} + ( - 4 \beta + 2) q^{28} + ( - 4 \beta - 2) q^{29} + (2 \beta - 4) q^{30} + (\beta + 3) q^{32} + ( - 6 \beta + 8) q^{34} - 2 q^{35} + ( - 10 \beta + 5) q^{36} + (4 \beta - 2) q^{37} + (8 \beta + 8) q^{39} + ( - \beta + 3) q^{40} - 6 q^{41} + ( - 4 \beta + 8) q^{42} + 6 q^{43} - 5 q^{45} + ( - 2 \beta + 4) q^{46} - 2 \beta q^{47} + 6 \beta q^{48} - 3 q^{49} + (\beta - 1) q^{50} + ( - 8 \beta + 8) q^{51} + ( - 6 \beta - 4) q^{52} + ( - 4 \beta + 6) q^{53} + ( - 4 \beta + 8) q^{54} + (2 \beta - 6) q^{56} + (2 \beta - 6) q^{58} + ( - 4 \beta - 4) q^{59} + ( - 2 \beta + 8) q^{60} + ( - 8 \beta - 2) q^{61} + 10 q^{63} + (2 \beta - 7) q^{64} + ( - 2 \beta - 4) q^{65} + ( - 6 \beta + 4) q^{67} + (10 \beta - 12) q^{68} + 8 q^{69} + ( - 2 \beta + 2) q^{70} - 8 \beta q^{71} + (5 \beta - 15) q^{72} + (2 \beta + 4) q^{73} + ( - 6 \beta + 10) q^{74} + 2 \beta q^{75} + 8 q^{78} - 4 q^{79} - 3 q^{80} + q^{81} + ( - 6 \beta + 6) q^{82} + 6 q^{83} + (4 \beta - 16) q^{84} + ( - 2 \beta + 4) q^{85} + (6 \beta - 6) q^{86} + ( - 4 \beta - 16) q^{87} + (8 \beta - 2) q^{89} + ( - 5 \beta + 5) q^{90} + (4 \beta + 8) q^{91} + (2 \beta - 8) q^{92} + (2 \beta - 4) q^{94} + (6 \beta + 4) q^{96} + ( - 4 \beta - 2) q^{97} + ( - 3 \beta + 3) q^{98} +O(q^{100})$$ q + (b - 1) * q^2 + 2*b * q^3 + (-2*b + 1) * q^4 - q^5 + (-2*b + 4) * q^6 + 2 * q^7 + (b - 3) * q^8 + 5 * q^9 + (-b + 1) * q^10 + (2*b - 8) * q^12 + (2*b + 4) * q^13 + (2*b - 2) * q^14 - 2*b * q^15 + 3 * q^16 + (2*b - 4) * q^17 + (5*b - 5) * q^18 + (2*b - 1) * q^20 + 4*b * q^21 + 2*b * q^23 + (-6*b + 4) * q^24 + q^25 + 2*b * q^26 + 4*b * q^27 + (-4*b + 2) * q^28 + (-4*b - 2) * q^29 + (2*b - 4) * q^30 + (b + 3) * q^32 + (-6*b + 8) * q^34 - 2 * q^35 + (-10*b + 5) * q^36 + (4*b - 2) * q^37 + (8*b + 8) * q^39 + (-b + 3) * q^40 - 6 * q^41 + (-4*b + 8) * q^42 + 6 * q^43 - 5 * q^45 + (-2*b + 4) * q^46 - 2*b * q^47 + 6*b * q^48 - 3 * q^49 + (b - 1) * q^50 + (-8*b + 8) * q^51 + (-6*b - 4) * q^52 + (-4*b + 6) * q^53 + (-4*b + 8) * q^54 + (2*b - 6) * q^56 + (2*b - 6) * q^58 + (-4*b - 4) * q^59 + (-2*b + 8) * q^60 + (-8*b - 2) * q^61 + 10 * q^63 + (2*b - 7) * q^64 + (-2*b - 4) * q^65 + (-6*b + 4) * q^67 + (10*b - 12) * q^68 + 8 * q^69 + (-2*b + 2) * q^70 - 8*b * q^71 + (5*b - 15) * q^72 + (2*b + 4) * q^73 + (-6*b + 10) * q^74 + 2*b * q^75 + 8 * q^78 - 4 * q^79 - 3 * q^80 + q^81 + (-6*b + 6) * q^82 + 6 * q^83 + (4*b - 16) * q^84 + (-2*b + 4) * q^85 + (6*b - 6) * q^86 + (-4*b - 16) * q^87 + (8*b - 2) * q^89 + (-5*b + 5) * q^90 + (4*b + 8) * q^91 + (2*b - 8) * q^92 + (2*b - 4) * q^94 + (6*b + 4) * q^96 + (-4*b - 2) * q^97 + (-3*b + 3) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 8 q^{6} + 4 q^{7} - 6 q^{8} + 10 q^{9}+O(q^{10})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 8 * q^6 + 4 * q^7 - 6 * q^8 + 10 * q^9 $$2 q - 2 q^{2} + 2 q^{4} - 2 q^{5} + 8 q^{6} + 4 q^{7} - 6 q^{8} + 10 q^{9} + 2 q^{10} - 16 q^{12} + 8 q^{13} - 4 q^{14} + 6 q^{16} - 8 q^{17} - 10 q^{18} - 2 q^{20} + 8 q^{24} + 2 q^{25} + 4 q^{28} - 4 q^{29} - 8 q^{30} + 6 q^{32} + 16 q^{34} - 4 q^{35} + 10 q^{36} - 4 q^{37} + 16 q^{39} + 6 q^{40} - 12 q^{41} + 16 q^{42} + 12 q^{43} - 10 q^{45} + 8 q^{46} - 6 q^{49} - 2 q^{50} + 16 q^{51} - 8 q^{52} + 12 q^{53} + 16 q^{54} - 12 q^{56} - 12 q^{58} - 8 q^{59} + 16 q^{60} - 4 q^{61} + 20 q^{63} - 14 q^{64} - 8 q^{65} + 8 q^{67} - 24 q^{68} + 16 q^{69} + 4 q^{70} - 30 q^{72} + 8 q^{73} + 20 q^{74} + 16 q^{78} - 8 q^{79} - 6 q^{80} + 2 q^{81} + 12 q^{82} + 12 q^{83} - 32 q^{84} + 8 q^{85} - 12 q^{86} - 32 q^{87} - 4 q^{89} + 10 q^{90} + 16 q^{91} - 16 q^{92} - 8 q^{94} + 8 q^{96} - 4 q^{97} + 6 q^{98}+O(q^{100})$$ 2 * q - 2 * q^2 + 2 * q^4 - 2 * q^5 + 8 * q^6 + 4 * q^7 - 6 * q^8 + 10 * q^9 + 2 * q^10 - 16 * q^12 + 8 * q^13 - 4 * q^14 + 6 * q^16 - 8 * q^17 - 10 * q^18 - 2 * q^20 + 8 * q^24 + 2 * q^25 + 4 * q^28 - 4 * q^29 - 8 * q^30 + 6 * q^32 + 16 * q^34 - 4 * q^35 + 10 * q^36 - 4 * q^37 + 16 * q^39 + 6 * q^40 - 12 * q^41 + 16 * q^42 + 12 * q^43 - 10 * q^45 + 8 * q^46 - 6 * q^49 - 2 * q^50 + 16 * q^51 - 8 * q^52 + 12 * q^53 + 16 * q^54 - 12 * q^56 - 12 * q^58 - 8 * q^59 + 16 * q^60 - 4 * q^61 + 20 * q^63 - 14 * q^64 - 8 * q^65 + 8 * q^67 - 24 * q^68 + 16 * q^69 + 4 * q^70 - 30 * q^72 + 8 * q^73 + 20 * q^74 + 16 * q^78 - 8 * q^79 - 6 * q^80 + 2 * q^81 + 12 * q^82 + 12 * q^83 - 32 * q^84 + 8 * q^85 - 12 * q^86 - 32 * q^87 - 4 * q^89 + 10 * q^90 + 16 * q^91 - 16 * q^92 - 8 * q^94 + 8 * q^96 - 4 * q^97 + 6 * q^98

## 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   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.41421 1.41421
−2.41421 −2.82843 3.82843 −1.00000 6.82843 2.00000 −4.41421 5.00000 2.41421
1.2 0.414214 2.82843 −1.82843 −1.00000 1.17157 2.00000 −1.58579 5.00000 −0.414214
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$5$$ $$1$$
$$11$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.a.d 2
3.b odd 2 1 5445.2.a.y 2
4.b odd 2 1 9680.2.a.bn 2
5.b even 2 1 3025.2.a.o 2
11.b odd 2 1 55.2.a.b 2
11.c even 5 4 605.2.g.l 8
11.d odd 10 4 605.2.g.f 8
33.d even 2 1 495.2.a.b 2
44.c even 2 1 880.2.a.m 2
55.d odd 2 1 275.2.a.c 2
55.e even 4 2 275.2.b.d 4
77.b even 2 1 2695.2.a.f 2
88.b odd 2 1 3520.2.a.bn 2
88.g even 2 1 3520.2.a.bo 2
132.d odd 2 1 7920.2.a.ch 2
143.d odd 2 1 9295.2.a.g 2
165.d even 2 1 2475.2.a.x 2
165.l odd 4 2 2475.2.c.l 4
220.g even 2 1 4400.2.a.bn 2
220.i odd 4 2 4400.2.b.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b 2 11.b odd 2 1
275.2.a.c 2 55.d odd 2 1
275.2.b.d 4 55.e even 4 2
495.2.a.b 2 33.d even 2 1
605.2.a.d 2 1.a even 1 1 trivial
605.2.g.f 8 11.d odd 10 4
605.2.g.l 8 11.c even 5 4
880.2.a.m 2 44.c even 2 1
2475.2.a.x 2 165.d even 2 1
2475.2.c.l 4 165.l odd 4 2
2695.2.a.f 2 77.b even 2 1
3025.2.a.o 2 5.b even 2 1
3520.2.a.bn 2 88.b odd 2 1
3520.2.a.bo 2 88.g even 2 1
4400.2.a.bn 2 220.g even 2 1
4400.2.b.q 4 220.i odd 4 2
5445.2.a.y 2 3.b odd 2 1
7920.2.a.ch 2 132.d odd 2 1
9295.2.a.g 2 143.d odd 2 1
9680.2.a.bn 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(605))$$:

 $$T_{2}^{2} + 2T_{2} - 1$$ T2^2 + 2*T2 - 1 $$T_{3}^{2} - 8$$ T3^2 - 8

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 2T - 1$$
$3$ $$T^{2} - 8$$
$5$ $$(T + 1)^{2}$$
$7$ $$(T - 2)^{2}$$
$11$ $$T^{2}$$
$13$ $$T^{2} - 8T + 8$$
$17$ $$T^{2} + 8T + 8$$
$19$ $$T^{2}$$
$23$ $$T^{2} - 8$$
$29$ $$T^{2} + 4T - 28$$
$31$ $$T^{2}$$
$37$ $$T^{2} + 4T - 28$$
$41$ $$(T + 6)^{2}$$
$43$ $$(T - 6)^{2}$$
$47$ $$T^{2} - 8$$
$53$ $$T^{2} - 12T + 4$$
$59$ $$T^{2} + 8T - 16$$
$61$ $$T^{2} + 4T - 124$$
$67$ $$T^{2} - 8T - 56$$
$71$ $$T^{2} - 128$$
$73$ $$T^{2} - 8T + 8$$
$79$ $$(T + 4)^{2}$$
$83$ $$(T - 6)^{2}$$
$89$ $$T^{2} + 4T - 124$$
$97$ $$T^{2} + 4T - 28$$