# Properties

 Label 60.5.g.b Level $60$ Weight $5$ Character orbit 60.g Analytic conductor $6.202$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$60 = 2^{2} \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 60.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.20219778503$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-5}, \sqrt{34})$$ Defining polynomial: $$x^{4} - 58 x^{2} + 1521$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$3\cdot 5$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -5 + \beta_{1} ) q^{3} -\beta_{3} q^{5} + ( -10 - \beta_{1} + \beta_{2} ) q^{7} + ( -14 - 10 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{9} +O(q^{10})$$ $$q + ( -5 + \beta_{1} ) q^{3} -\beta_{3} q^{5} + ( -10 - \beta_{1} + \beta_{2} ) q^{7} + ( -14 - 10 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{9} + ( -10 \beta_{1} - 2 \beta_{2} - 12 \beta_{3} ) q^{11} + ( -10 - 16 \beta_{1} + 16 \beta_{2} ) q^{13} + ( -25 + 5 \beta_{2} + 6 \beta_{3} ) q^{15} + ( -20 \beta_{1} - 4 \beta_{2} - 30 \beta_{3} ) q^{17} + ( -136 + 10 \beta_{1} - 10 \beta_{2} ) q^{19} + ( -1 - 10 \beta_{1} - 6 \beta_{2} + 9 \beta_{3} ) q^{21} + ( 85 \beta_{1} + 17 \beta_{2} ) q^{23} -125 q^{25} + ( 205 + 26 \beta_{1} - 13 \beta_{2} + 60 \beta_{3} ) q^{27} + ( -80 \beta_{1} - 16 \beta_{2} - 36 \beta_{3} ) q^{29} + ( 584 + 40 \beta_{1} - 40 \beta_{2} ) q^{31} + ( 270 + 60 \beta_{1} + 48 \beta_{2} + 90 \beta_{3} ) q^{33} + ( -25 \beta_{1} - 5 \beta_{2} + 4 \beta_{3} ) q^{35} + ( -70 + 12 \beta_{1} - 12 \beta_{2} ) q^{37} + ( -766 - 10 \beta_{1} - 96 \beta_{2} + 144 \beta_{3} ) q^{39} + ( 120 \beta_{1} + 24 \beta_{2} - 66 \beta_{3} ) q^{41} + ( 1850 + 29 \beta_{1} - 29 \beta_{2} ) q^{43} + ( -175 - 50 \beta_{1} - 50 \beta_{2} - 6 \beta_{3} ) q^{45} + ( 185 \beta_{1} + 37 \beta_{2} ) q^{47} + ( -1995 + 20 \beta_{1} - 20 \beta_{2} ) q^{49} + ( 390 + 120 \beta_{1} + 126 \beta_{2} + 216 \beta_{3} ) q^{51} + ( 140 \beta_{1} + 28 \beta_{2} - 270 \beta_{3} ) q^{53} + ( -1200 + 50 \beta_{1} - 50 \beta_{2} ) q^{55} + ( 1190 - 136 \beta_{1} + 60 \beta_{2} - 90 \beta_{3} ) q^{57} + ( -740 \beta_{1} - 148 \beta_{2} - 228 \beta_{3} ) q^{59} + ( -16 - 300 \beta_{1} + 300 \beta_{2} ) q^{61} + ( 1160 + 79 \beta_{1} - 41 \beta_{2} - 60 \beta_{3} ) q^{63} + ( -400 \beta_{1} - 80 \beta_{2} - 86 \beta_{3} ) q^{65} + ( -2230 - 171 \beta_{1} + 171 \beta_{2} ) q^{67} + ( -4845 - 510 \beta_{1} + 102 \beta_{2} - 153 \beta_{3} ) q^{69} + ( 510 \beta_{1} + 102 \beta_{2} - 48 \beta_{3} ) q^{71} + ( 4610 - 228 \beta_{1} + 228 \beta_{2} ) q^{73} + ( 625 - 125 \beta_{1} ) q^{75} + ( -140 \beta_{1} - 28 \beta_{2} - 60 \beta_{3} ) q^{77} + ( -1828 + 390 \beta_{1} - 390 \beta_{2} ) q^{79} + ( 631 + 140 \beta_{1} - 196 \beta_{2} - 516 \beta_{3} ) q^{81} + ( 745 \beta_{1} + 149 \beta_{2} + 360 \beta_{3} ) q^{83} + ( -3150 + 100 \beta_{1} - 100 \beta_{2} ) q^{85} + ( 3660 + 480 \beta_{1} + 84 \beta_{2} + 360 \beta_{3} ) q^{87} + ( -120 \beta_{1} - 24 \beta_{2} + 876 \beta_{3} ) q^{89} + ( 4996 + 170 \beta_{1} - 170 \beta_{2} ) q^{91} + ( -880 + 584 \beta_{1} + 240 \beta_{2} - 360 \beta_{3} ) q^{93} + ( 250 \beta_{1} + 50 \beta_{2} + 196 \beta_{3} ) q^{95} + ( -2470 + 484 \beta_{1} - 484 \beta_{2} ) q^{97} + ( -5760 - 270 \beta_{1} - 522 \beta_{2} - 432 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 20q^{3} - 40q^{7} - 56q^{9} + O(q^{10})$$ $$4q - 20q^{3} - 40q^{7} - 56q^{9} - 40q^{13} - 100q^{15} - 544q^{19} - 4q^{21} - 500q^{25} + 820q^{27} + 2336q^{31} + 1080q^{33} - 280q^{37} - 3064q^{39} + 7400q^{43} - 700q^{45} - 7980q^{49} + 1560q^{51} - 4800q^{55} + 4760q^{57} - 64q^{61} + 4640q^{63} - 8920q^{67} - 19380q^{69} + 18440q^{73} + 2500q^{75} - 7312q^{79} + 2524q^{81} - 12600q^{85} + 14640q^{87} + 19984q^{91} - 3520q^{93} - 9880q^{97} - 23040q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 58 x^{2} + 1521$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 13 \nu^{2} - 45 \nu + 377$$$$)/52$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} - 13 \nu^{2} + 149 \nu + 377$$$$)/52$$ $$\beta_{3}$$ $$=$$ $$($$$$-5 \nu^{3} + 95 \nu$$$$)/78$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-3 \beta_{3} + 5 \beta_{2} - 5 \beta_{1}$$$$)/15$$ $$\nu^{2}$$ $$=$$ $$($$$$-12 \beta_{3} - 10 \beta_{2} - 50 \beta_{1} + 435$$$$)/15$$ $$\nu^{3}$$ $$=$$ $$($$$$-291 \beta_{3} + 95 \beta_{2} - 95 \beta_{1}$$$$)/15$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/60\mathbb{Z}\right)^\times$$.

 $$n$$ $$31$$ $$37$$ $$41$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1
 5.83095 + 2.23607i 5.83095 − 2.23607i −5.83095 − 2.23607i −5.83095 + 2.23607i
0 −7.91548 4.28313i 0 11.1803i 0 7.49286 0 44.3095 + 67.8061i 0
41.2 0 −7.91548 + 4.28313i 0 11.1803i 0 7.49286 0 44.3095 67.8061i 0
41.3 0 −2.08452 8.75527i 0 11.1803i 0 −27.4929 0 −72.3095 + 36.5011i 0
41.4 0 −2.08452 + 8.75527i 0 11.1803i 0 −27.4929 0 −72.3095 36.5011i 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
3.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 60.5.g.b 4
3.b odd 2 1 inner 60.5.g.b 4
4.b odd 2 1 240.5.l.c 4
5.b even 2 1 300.5.g.g 4
5.c odd 4 2 300.5.b.d 8
12.b even 2 1 240.5.l.c 4
15.d odd 2 1 300.5.g.g 4
15.e even 4 2 300.5.b.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.5.g.b 4 1.a even 1 1 trivial
60.5.g.b 4 3.b odd 2 1 inner
240.5.l.c 4 4.b odd 2 1
240.5.l.c 4 12.b even 2 1
300.5.b.d 8 5.c odd 4 2
300.5.b.d 8 15.e even 4 2
300.5.g.g 4 5.b even 2 1
300.5.g.g 4 15.d odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 20 T_{7} - 206$$ acting on $$S_{5}^{\mathrm{new}}(60, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$6561 + 1620 T + 228 T^{2} + 20 T^{3} + T^{4}$$
$5$ $$( 125 + T^{2} )^{2}$$
$7$ $$( -206 + 20 T + T^{2} )^{2}$$
$11$ $$29160000 + 35280 T^{2} + T^{4}$$
$13$ $$( -78236 + 20 T + T^{2} )^{2}$$
$17$ $$3014010000 + 207720 T^{2} + T^{4}$$
$19$ $$( -12104 + 272 T + T^{2} )^{2}$$
$23$ $$152217022500 + 988380 T^{2} + T^{4}$$
$29$ $$127020960000 + 853920 T^{2} + T^{4}$$
$31$ $$( -148544 - 1168 T + T^{2} )^{2}$$
$37$ $$( -39164 + 140 T + T^{2} )^{2}$$
$41$ $$58612410000 + 4009320 T^{2} + T^{4}$$
$43$ $$( 3165154 - 3700 T + T^{2} )^{2}$$
$47$ $$3415658422500 + 4681980 T^{2} + T^{4}$$
$53$ $$106545748410000 + 25442280 T^{2} + T^{4}$$
$59$ $$1101947859360000 + 67661280 T^{2} + T^{4}$$
$61$ $$( -27539744 + 32 T + T^{2} )^{2}$$
$67$ $$( -3974846 + 4460 T + T^{2} )^{2}$$
$71$ $$151009689960000 + 39095280 T^{2} + T^{4}$$
$73$ $$( 5344996 - 9220 T + T^{2} )^{2}$$
$79$ $$( -43201016 + 3656 T + T^{2} )^{2}$$
$83$ $$891819673222500 + 76143420 T^{2} + T^{4}$$
$89$ $$10292427142560000 + 206428320 T^{2} + T^{4}$$
$97$ $$( -65581436 + 4940 T + T^{2} )^{2}$$