# Properties

 Label 605.2.a.k Level $605$ Weight $2$ Character orbit 605.a Self dual yes Analytic conductor $4.831$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.725.1 Defining polynomial: $$x^{4} - x^{3} - 3 x^{2} + x + 1$$ 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

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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{2} + \beta_{3} ) q^{3} + ( -1 + \beta_{1} + \beta_{2} ) q^{4} - q^{5} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{6} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{7} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{8} + ( 1 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( -1 + \beta_{2} + \beta_{3} ) q^{3} + ( -1 + \beta_{1} + \beta_{2} ) q^{4} - q^{5} + ( -1 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{6} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{7} + ( -\beta_{1} + \beta_{2} + \beta_{3} ) q^{8} + ( 1 + 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{9} -\beta_{1} q^{10} + ( 2 + 2 \beta_{1} - \beta_{3} ) q^{12} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{13} + ( -1 + 2 \beta_{1} + 2 \beta_{3} ) q^{14} + ( 1 - \beta_{2} - \beta_{3} ) q^{15} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{16} + ( -1 - \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{17} + ( 3 - \beta_{3} ) q^{18} + ( 5 + 2 \beta_{1} - \beta_{2} ) q^{19} + ( 1 - \beta_{1} - \beta_{2} ) q^{20} + ( 2 + 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{21} + ( 2 - \beta_{1} - \beta_{2} ) q^{23} + ( 4 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{24} + q^{25} + ( -3 - \beta_{1} - \beta_{2} ) q^{26} + ( -4 - \beta_{1} + \beta_{3} ) q^{27} + ( 2 + \beta_{1} + 2 \beta_{3} ) q^{28} + ( 5 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{29} + ( 1 - \beta_{1} - \beta_{2} - \beta_{3} ) q^{30} -5 \beta_{1} q^{31} + ( 1 - 2 \beta_{2} - 4 \beta_{3} ) q^{32} + ( 2 \beta_{2} - \beta_{3} ) q^{34} + ( -\beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{35} + ( -2 - 2 \beta_{1} + \beta_{2} + 4 \beta_{3} ) q^{36} + ( 3 + 3 \beta_{1} + 3 \beta_{2} - 7 \beta_{3} ) q^{37} + ( 3 + 6 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{38} + ( 5 - \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{39} + ( \beta_{1} - \beta_{2} - \beta_{3} ) q^{40} + ( 1 - \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{41} + ( 1 + 4 \beta_{1} + \beta_{2} + \beta_{3} ) q^{42} + ( 6 - \beta_{1} - 2 \beta_{2} ) q^{43} + ( -1 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{45} + ( -\beta_{2} - \beta_{3} ) q^{46} + ( 3 - \beta_{1} - 3 \beta_{2} ) q^{47} + ( -2 - 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{48} + ( -1 - \beta_{1} + 3 \beta_{2} + \beta_{3} ) q^{49} + \beta_{1} q^{50} + ( 3 + \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{51} + ( \beta_{1} - \beta_{2} - 5 \beta_{3} ) q^{52} + ( -6 + \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{53} + ( -1 - 4 \beta_{1} ) q^{54} + ( 3 + \beta_{1} + 3 \beta_{2} - 4 \beta_{3} ) q^{56} + ( -9 + \beta_{1} + 7 \beta_{2} + 8 \beta_{3} ) q^{57} + ( -1 - 4 \beta_{2} - \beta_{3} ) q^{58} + ( 2 - \beta_{1} - 4 \beta_{2} + 5 \beta_{3} ) q^{59} + ( -2 - 2 \beta_{1} + \beta_{3} ) q^{60} + ( 2 - 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{61} + ( -5 - 5 \beta_{1} - 5 \beta_{2} ) q^{62} + ( -3 - \beta_{1} - 2 \beta_{2} + 6 \beta_{3} ) q^{63} + ( 2 - 3 \beta_{1} - 4 \beta_{3} ) q^{64} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{65} + ( -5 - 3 \beta_{1} - 4 \beta_{2} + 6 \beta_{3} ) q^{67} + ( 3 \beta_{1} + \beta_{2} - 4 \beta_{3} ) q^{68} + ( -3 - 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{69} + ( 1 - 2 \beta_{1} - 2 \beta_{3} ) q^{70} + ( 3 - 3 \beta_{1} - 2 \beta_{3} ) q^{71} + ( -9 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{72} + ( 1 + 3 \beta_{1} + 2 \beta_{2} ) q^{73} + ( 2 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{74} + ( -1 + \beta_{2} + \beta_{3} ) q^{75} + ( -6 + 6 \beta_{1} + 7 \beta_{2} + 2 \beta_{3} ) q^{76} + ( 2 - 2 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} ) q^{78} + ( 9 + 4 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} ) q^{79} + ( \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{80} + ( 3 - 6 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{81} + ( -4 + 4 \beta_{1} + 3 \beta_{3} ) q^{82} + ( -1 - \beta_{1} + 3 \beta_{2} - 6 \beta_{3} ) q^{83} + ( -1 + 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} ) q^{84} + ( 1 + \beta_{1} + \beta_{2} - 3 \beta_{3} ) q^{85} + ( 1 + 3 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{86} + ( -7 - 5 \beta_{1} + 3 \beta_{2} + 4 \beta_{3} ) q^{87} + ( 1 - 6 \beta_{2} ) q^{89} + ( -3 + \beta_{3} ) q^{90} + ( 1 + 2 \beta_{2} - 8 \beta_{3} ) q^{91} + ( -3 + \beta_{2} - \beta_{3} ) q^{92} + ( 5 - 5 \beta_{1} - 5 \beta_{2} - 5 \beta_{3} ) q^{93} + ( 2 - \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{94} + ( -5 - 2 \beta_{1} + \beta_{2} ) q^{95} + ( -9 - 6 \beta_{1} + \beta_{2} + 3 \beta_{3} ) q^{96} + ( 5 - 2 \beta_{2} + 8 \beta_{3} ) q^{97} + ( -4 + 2 \beta_{1} + 3 \beta_{3} ) q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + q^{2} - q^{4} - 4q^{5} + q^{6} + 3q^{7} + 3q^{8} + O(q^{10})$$ $$4q + q^{2} - q^{4} - 4q^{5} + q^{6} + 3q^{7} + 3q^{8} - q^{10} + 8q^{12} + q^{13} + 2q^{14} - 3q^{16} - q^{17} + 10q^{18} + 20q^{19} + q^{20} + 10q^{21} + 5q^{23} + 11q^{24} + 4q^{25} - 15q^{26} - 15q^{27} + 13q^{28} + 12q^{29} - q^{30} - 5q^{31} - 8q^{32} + 2q^{34} - 3q^{35} + 7q^{37} + 20q^{38} + 7q^{39} - 3q^{40} + 11q^{41} + 12q^{42} + 19q^{43} - 4q^{46} + 5q^{47} - 10q^{48} + 3q^{49} + q^{50} + 7q^{51} - 11q^{52} - 11q^{53} - 8q^{54} + 11q^{56} - 5q^{57} - 14q^{58} + 9q^{59} - 8q^{60} + 12q^{61} - 35q^{62} - 5q^{63} - 3q^{64} - q^{65} - 19q^{67} - 3q^{68} - 8q^{69} - 2q^{70} + 5q^{71} - 25q^{72} + 11q^{73} - 8q^{78} + 34q^{79} + 3q^{80} + 4q^{81} - 6q^{82} - 11q^{83} + 11q^{84} + q^{85} + q^{86} - 19q^{87} - 8q^{89} - 10q^{90} - 8q^{91} - 12q^{92} - 5q^{93} - q^{94} - 20q^{95} - 34q^{96} + 32q^{97} - 8q^{98} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 3 x^{2} + x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 2 \nu + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 1$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 3 \beta_{1}$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 −1.35567 −0.477260 0.737640 2.09529
−1.35567 0.575493 −0.162147 −1.00000 −0.780181 3.64941 2.93117 −2.66881 1.35567
1.2 −0.477260 0.323071 −1.77222 −1.00000 −0.154189 −2.68522 1.80033 −2.89563 0.477260
1.3 0.737640 −2.81156 −1.45589 −1.00000 −2.07392 −1.03138 −2.54920 4.90488 −0.737640
1.4 2.09529 1.91300 2.39026 −1.00000 4.00829 3.06719 0.817703 0.659557 −2.09529
 $$n$$: 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.k 4
3.b odd 2 1 5445.2.a.bi 4
4.b odd 2 1 9680.2.a.cm 4
5.b even 2 1 3025.2.a.w 4
11.b odd 2 1 605.2.a.j 4
11.c even 5 2 605.2.g.e 8
11.c even 5 2 605.2.g.k 8
11.d odd 10 2 55.2.g.b 8
11.d odd 10 2 605.2.g.m 8
33.d even 2 1 5445.2.a.bp 4
33.f even 10 2 495.2.n.e 8
44.c even 2 1 9680.2.a.cn 4
44.g even 10 2 880.2.bo.h 8
55.d odd 2 1 3025.2.a.bd 4
55.h odd 10 2 275.2.h.a 8
55.l even 20 4 275.2.z.a 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.g.b 8 11.d odd 10 2
275.2.h.a 8 55.h odd 10 2
275.2.z.a 16 55.l even 20 4
495.2.n.e 8 33.f even 10 2
605.2.a.j 4 11.b odd 2 1
605.2.a.k 4 1.a even 1 1 trivial
605.2.g.e 8 11.c even 5 2
605.2.g.k 8 11.c even 5 2
605.2.g.m 8 11.d odd 10 2
880.2.bo.h 8 44.g even 10 2
3025.2.a.w 4 5.b even 2 1
3025.2.a.bd 4 55.d odd 2 1
5445.2.a.bi 4 3.b odd 2 1
5445.2.a.bp 4 33.d even 2 1
9680.2.a.cm 4 4.b odd 2 1
9680.2.a.cn 4 44.c even 2 1

## Hecke kernels

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

 $$T_{2}^{4} - T_{2}^{3} - 3 T_{2}^{2} + T_{2} + 1$$ $$T_{3}^{4} - 6 T_{3}^{2} + 5 T_{3} - 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T - 3 T^{2} - T^{3} + T^{4}$$
$3$ $$-1 + 5 T - 6 T^{2} + T^{4}$$
$5$ $$( 1 + T )^{4}$$
$7$ $$31 + 23 T - 11 T^{2} - 3 T^{3} + T^{4}$$
$11$ $$T^{4}$$
$13$ $$139 + 7 T - 25 T^{2} - T^{3} + T^{4}$$
$17$ $$19 - 32 T - 20 T^{2} + T^{3} + T^{4}$$
$19$ $$25 - 275 T + 130 T^{2} - 20 T^{3} + T^{4}$$
$23$ $$-11 + 10 T + 4 T^{2} - 5 T^{3} + T^{4}$$
$29$ $$-451 + 171 T + 20 T^{2} - 12 T^{3} + T^{4}$$
$31$ $$625 - 125 T - 75 T^{2} + 5 T^{3} + T^{4}$$
$37$ $$-1151 + 826 T - 100 T^{2} - 7 T^{3} + T^{4}$$
$41$ $$-319 + 174 T + 6 T^{2} - 11 T^{3} + T^{4}$$
$43$ $$211 - 289 T + 121 T^{2} - 19 T^{3} + T^{4}$$
$47$ $$169 + 65 T - 21 T^{2} - 5 T^{3} + T^{4}$$
$53$ $$941 - 311 T - 43 T^{2} + 11 T^{3} + T^{4}$$
$59$ $$-829 + 549 T - 73 T^{2} - 9 T^{3} + T^{4}$$
$61$ $$-169 + 78 T + 23 T^{2} - 12 T^{3} + T^{4}$$
$67$ $$-4079 - 1014 T + 22 T^{2} + 19 T^{3} + T^{4}$$
$71$ $$-131 + 170 T - 46 T^{2} - 5 T^{3} + T^{4}$$
$73$ $$-11 + 12 T + 10 T^{2} - 11 T^{3} + T^{4}$$
$79$ $$-6779 - 299 T + 336 T^{2} - 34 T^{3} + T^{4}$$
$83$ $$-1699 - 1239 T - 99 T^{2} + 11 T^{3} + T^{4}$$
$89$ $$1861 - 472 T - 102 T^{2} + 8 T^{3} + T^{4}$$
$97$ $$-3011 + 896 T + 210 T^{2} - 32 T^{3} + T^{4}$$