# Properties

 Label 105.4.q.a Level $105$ Weight $4$ Character orbit 105.q Analytic conductor $6.195$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$105 = 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 105.q (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.19520055060$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + 5 \zeta_{12} q^{2} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{3} + 17 \zeta_{12}^{2} q^{4} + ( -2 + 11 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{5} -15 q^{6} + ( -18 \zeta_{12} - \zeta_{12}^{3} ) q^{7} + 45 \zeta_{12}^{3} q^{8} + ( 9 - 9 \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + 5 \zeta_{12} q^{2} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{3} + 17 \zeta_{12}^{2} q^{4} + ( -2 + 11 \zeta_{12} + 2 \zeta_{12}^{2} ) q^{5} -15 q^{6} + ( -18 \zeta_{12} - \zeta_{12}^{3} ) q^{7} + 45 \zeta_{12}^{3} q^{8} + ( 9 - 9 \zeta_{12}^{2} ) q^{9} + ( -10 \zeta_{12} + 55 \zeta_{12}^{2} + 10 \zeta_{12}^{3} ) q^{10} + 45 \zeta_{12}^{2} q^{11} -51 \zeta_{12} q^{12} -37 \zeta_{12}^{3} q^{13} + ( 5 - 95 \zeta_{12}^{2} ) q^{14} + ( -33 - 6 \zeta_{12}^{3} ) q^{15} + ( -89 + 89 \zeta_{12}^{2} ) q^{16} + ( -58 \zeta_{12} + 58 \zeta_{12}^{3} ) q^{17} + ( 45 \zeta_{12} - 45 \zeta_{12}^{3} ) q^{18} + ( 135 - 135 \zeta_{12}^{2} ) q^{19} + ( -34 + 187 \zeta_{12}^{3} ) q^{20} + ( 54 + 3 \zeta_{12}^{2} ) q^{21} + 225 \zeta_{12}^{3} q^{22} + 63 \zeta_{12} q^{23} -135 \zeta_{12}^{2} q^{24} + ( -44 \zeta_{12} + 117 \zeta_{12}^{2} + 44 \zeta_{12}^{3} ) q^{25} + ( 185 - 185 \zeta_{12}^{2} ) q^{26} + 27 \zeta_{12}^{3} q^{27} + ( 17 \zeta_{12} - 323 \zeta_{12}^{3} ) q^{28} + 184 q^{29} + ( 30 - 165 \zeta_{12} - 30 \zeta_{12}^{2} ) q^{30} -170 \zeta_{12}^{2} q^{31} + ( -85 \zeta_{12} + 85 \zeta_{12}^{3} ) q^{32} -135 \zeta_{12} q^{33} -290 q^{34} + ( 11 + 38 \zeta_{12} - 209 \zeta_{12}^{2} - 36 \zeta_{12}^{3} ) q^{35} + 153 q^{36} -71 \zeta_{12} q^{37} + ( 675 \zeta_{12} - 675 \zeta_{12}^{3} ) q^{38} + 111 \zeta_{12}^{2} q^{39} + ( -495 - 90 \zeta_{12} + 495 \zeta_{12}^{2} ) q^{40} + 247 q^{41} + ( 270 \zeta_{12} + 15 \zeta_{12}^{3} ) q^{42} -242 \zeta_{12}^{3} q^{43} + ( -765 + 765 \zeta_{12}^{2} ) q^{44} + ( 99 \zeta_{12} + 18 \zeta_{12}^{2} - 99 \zeta_{12}^{3} ) q^{45} + 315 \zeta_{12}^{2} q^{46} -567 \zeta_{12} q^{47} -267 \zeta_{12}^{3} q^{48} + ( -37 + 360 \zeta_{12}^{2} ) q^{49} + ( -220 + 585 \zeta_{12}^{3} ) q^{50} + ( 174 - 174 \zeta_{12}^{2} ) q^{51} + ( 629 \zeta_{12} - 629 \zeta_{12}^{3} ) q^{52} + ( 45 \zeta_{12} - 45 \zeta_{12}^{3} ) q^{53} + ( -135 + 135 \zeta_{12}^{2} ) q^{54} + ( -90 + 495 \zeta_{12}^{3} ) q^{55} + ( 855 - 810 \zeta_{12}^{2} ) q^{56} + 405 \zeta_{12}^{3} q^{57} + 920 \zeta_{12} q^{58} -4 \zeta_{12}^{2} q^{59} + ( 102 \zeta_{12} - 561 \zeta_{12}^{2} - 102 \zeta_{12}^{3} ) q^{60} + ( -262 + 262 \zeta_{12}^{2} ) q^{61} -850 \zeta_{12}^{3} q^{62} + ( -171 \zeta_{12} + 162 \zeta_{12}^{3} ) q^{63} + 287 q^{64} + ( 407 + 74 \zeta_{12} - 407 \zeta_{12}^{2} ) q^{65} -675 \zeta_{12}^{2} q^{66} + ( -874 \zeta_{12} + 874 \zeta_{12}^{3} ) q^{67} -986 \zeta_{12} q^{68} -189 q^{69} + ( 180 + 55 \zeta_{12} + 10 \zeta_{12}^{2} - 1045 \zeta_{12}^{3} ) q^{70} -1138 q^{71} + 405 \zeta_{12} q^{72} + ( -108 \zeta_{12} + 108 \zeta_{12}^{3} ) q^{73} -355 \zeta_{12}^{2} q^{74} + ( 132 - 351 \zeta_{12} - 132 \zeta_{12}^{2} ) q^{75} + 2295 q^{76} + ( 45 \zeta_{12} - 855 \zeta_{12}^{3} ) q^{77} + 555 \zeta_{12}^{3} q^{78} + ( -62 + 62 \zeta_{12}^{2} ) q^{79} + ( -979 \zeta_{12} - 178 \zeta_{12}^{2} + 979 \zeta_{12}^{3} ) q^{80} -81 \zeta_{12}^{2} q^{81} + 1235 \zeta_{12} q^{82} -454 \zeta_{12}^{3} q^{83} + ( -51 + 969 \zeta_{12}^{2} ) q^{84} + ( -638 - 116 \zeta_{12}^{3} ) q^{85} + ( 1210 - 1210 \zeta_{12}^{2} ) q^{86} + ( -552 \zeta_{12} + 552 \zeta_{12}^{3} ) q^{87} + ( -2025 \zeta_{12} + 2025 \zeta_{12}^{3} ) q^{88} + ( -1206 + 1206 \zeta_{12}^{2} ) q^{89} + ( 495 + 90 \zeta_{12}^{3} ) q^{90} + ( -703 + 666 \zeta_{12}^{2} ) q^{91} + 1071 \zeta_{12}^{3} q^{92} + 510 \zeta_{12} q^{93} -2835 \zeta_{12}^{2} q^{94} + ( 1485 \zeta_{12} + 270 \zeta_{12}^{2} - 1485 \zeta_{12}^{3} ) q^{95} + ( 255 - 255 \zeta_{12}^{2} ) q^{96} + 1360 \zeta_{12}^{3} q^{97} + ( -185 \zeta_{12} + 1800 \zeta_{12}^{3} ) q^{98} + 405 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 34q^{4} - 4q^{5} - 60q^{6} + 18q^{9} + O(q^{10})$$ $$4q + 34q^{4} - 4q^{5} - 60q^{6} + 18q^{9} + 110q^{10} + 90q^{11} - 170q^{14} - 132q^{15} - 178q^{16} + 270q^{19} - 136q^{20} + 222q^{21} - 270q^{24} + 234q^{25} + 370q^{26} + 736q^{29} + 60q^{30} - 340q^{31} - 1160q^{34} - 374q^{35} + 612q^{36} + 222q^{39} - 990q^{40} + 988q^{41} - 1530q^{44} + 36q^{45} + 630q^{46} + 572q^{49} - 880q^{50} + 348q^{51} - 270q^{54} - 360q^{55} + 1800q^{56} - 8q^{59} - 1122q^{60} - 524q^{61} + 1148q^{64} + 814q^{65} - 1350q^{66} - 756q^{69} + 740q^{70} - 4552q^{71} - 710q^{74} + 264q^{75} + 9180q^{76} - 124q^{79} - 356q^{80} - 162q^{81} + 1734q^{84} - 2552q^{85} + 2420q^{86} - 2412q^{89} + 1980q^{90} - 1480q^{91} - 5670q^{94} + 540q^{95} + 510q^{96} + 1620q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/105\mathbb{Z}\right)^\times$$.

 $$n$$ $$22$$ $$31$$ $$71$$ $$\chi(n)$$ $$-1$$ $$-1 + \zeta_{12}^{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
4.1
 −0.866025 + 0.500000i 0.866025 − 0.500000i −0.866025 − 0.500000i 0.866025 + 0.500000i
−4.33013 + 2.50000i 2.59808 + 1.50000i 8.50000 14.7224i −10.5263 + 3.76795i −15.0000 15.5885 10.0000i 45.0000i 4.50000 + 7.79423i 36.1603 42.6314i
4.2 4.33013 2.50000i −2.59808 1.50000i 8.50000 14.7224i 8.52628 7.23205i −15.0000 −15.5885 + 10.0000i 45.0000i 4.50000 + 7.79423i 18.8397 52.6314i
79.1 −4.33013 2.50000i 2.59808 1.50000i 8.50000 + 14.7224i −10.5263 3.76795i −15.0000 15.5885 + 10.0000i 45.0000i 4.50000 7.79423i 36.1603 + 42.6314i
79.2 4.33013 + 2.50000i −2.59808 + 1.50000i 8.50000 + 14.7224i 8.52628 + 7.23205i −15.0000 −15.5885 10.0000i 45.0000i 4.50000 7.79423i 18.8397 + 52.6314i
 $$n$$: 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
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.4.q.a 4
5.b even 2 1 inner 105.4.q.a 4
7.c even 3 1 inner 105.4.q.a 4
35.j even 6 1 inner 105.4.q.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.q.a 4 1.a even 1 1 trivial
105.4.q.a 4 5.b even 2 1 inner
105.4.q.a 4 7.c even 3 1 inner
105.4.q.a 4 35.j even 6 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 25 T_{2}^{2} + 625$$ acting on $$S_{4}^{\mathrm{new}}(105, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$625 - 25 T^{2} + T^{4}$$
$3$ $$81 - 9 T^{2} + T^{4}$$
$5$ $$15625 + 500 T - 109 T^{2} + 4 T^{3} + T^{4}$$
$7$ $$117649 - 286 T^{2} + T^{4}$$
$11$ $$( 2025 - 45 T + T^{2} )^{2}$$
$13$ $$( 1369 + T^{2} )^{2}$$
$17$ $$11316496 - 3364 T^{2} + T^{4}$$
$19$ $$( 18225 - 135 T + T^{2} )^{2}$$
$23$ $$15752961 - 3969 T^{2} + T^{4}$$
$29$ $$( -184 + T )^{4}$$
$31$ $$( 28900 + 170 T + T^{2} )^{2}$$
$37$ $$25411681 - 5041 T^{2} + T^{4}$$
$41$ $$( -247 + T )^{4}$$
$43$ $$( 58564 + T^{2} )^{2}$$
$47$ $$103355177121 - 321489 T^{2} + T^{4}$$
$53$ $$4100625 - 2025 T^{2} + T^{4}$$
$59$ $$( 16 + 4 T + T^{2} )^{2}$$
$61$ $$( 68644 + 262 T + T^{2} )^{2}$$
$67$ $$583506543376 - 763876 T^{2} + T^{4}$$
$71$ $$( 1138 + T )^{4}$$
$73$ $$136048896 - 11664 T^{2} + T^{4}$$
$79$ $$( 3844 + 62 T + T^{2} )^{2}$$
$83$ $$( 206116 + T^{2} )^{2}$$
$89$ $$( 1454436 + 1206 T + T^{2} )^{2}$$
$97$ $$( 1849600 + T^{2} )^{2}$$