# Properties

 Label 105.3.k.a Level 105 Weight 3 Character orbit 105.k Analytic conductor 2.861 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$2.86104277578$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{8} q^{2} + ( -2 - \zeta_{8} + 2 \zeta_{8}^{2} ) q^{3} -3 \zeta_{8}^{2} q^{4} + ( -4 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{5} + ( -2 \zeta_{8} - \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{6} -7 q^{7} -7 \zeta_{8}^{3} q^{8} + ( 4 \zeta_{8} - 7 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + \zeta_{8} q^{2} + ( -2 - \zeta_{8} + 2 \zeta_{8}^{2} ) q^{3} -3 \zeta_{8}^{2} q^{4} + ( -4 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{5} + ( -2 \zeta_{8} - \zeta_{8}^{2} + 2 \zeta_{8}^{3} ) q^{6} -7 q^{7} -7 \zeta_{8}^{3} q^{8} + ( 4 \zeta_{8} - 7 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{9} + ( 3 - 4 \zeta_{8}^{2} ) q^{10} + ( -7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{11} + ( 6 + 6 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{12} + ( -8 + 8 \zeta_{8}^{2} ) q^{13} -7 \zeta_{8} q^{14} + ( -3 + 14 \zeta_{8} + 4 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{15} -5 q^{16} -26 \zeta_{8}^{3} q^{17} + ( 4 + 4 \zeta_{8}^{2} - 7 \zeta_{8}^{3} ) q^{18} + 10 q^{19} + ( -9 \zeta_{8} + 12 \zeta_{8}^{3} ) q^{20} + ( 14 + 7 \zeta_{8} - 14 \zeta_{8}^{2} ) q^{21} + ( 7 - 7 \zeta_{8}^{2} ) q^{22} + 34 \zeta_{8}^{3} q^{23} + ( -7 + 14 \zeta_{8} + 14 \zeta_{8}^{3} ) q^{24} + ( -24 + 7 \zeta_{8}^{2} ) q^{25} + ( -8 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{26} + ( 10 + 10 \zeta_{8}^{2} + 23 \zeta_{8}^{3} ) q^{27} + 21 \zeta_{8}^{2} q^{28} + ( -3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{29} + ( 2 - 3 \zeta_{8} + 14 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{30} -14 \zeta_{8}^{2} q^{31} -33 \zeta_{8} q^{32} + ( -7 + 28 \zeta_{8} + 7 \zeta_{8}^{2} ) q^{33} + 26 q^{34} + ( 28 \zeta_{8} + 21 \zeta_{8}^{3} ) q^{35} + ( -21 - 12 \zeta_{8} - 12 \zeta_{8}^{3} ) q^{36} + ( 30 - 30 \zeta_{8}^{2} ) q^{37} + 10 \zeta_{8} q^{38} + ( 8 \zeta_{8} - 32 \zeta_{8}^{2} - 8 \zeta_{8}^{3} ) q^{39} + ( -28 - 21 \zeta_{8}^{2} ) q^{40} + ( 24 \zeta_{8} - 24 \zeta_{8}^{3} ) q^{41} + ( 14 \zeta_{8} + 7 \zeta_{8}^{2} - 14 \zeta_{8}^{3} ) q^{42} + ( 36 + 36 \zeta_{8}^{2} ) q^{43} + ( -21 \zeta_{8} + 21 \zeta_{8}^{3} ) q^{44} + ( -4 - 21 \zeta_{8} - 28 \zeta_{8}^{2} + 28 \zeta_{8}^{3} ) q^{45} -34 q^{46} -6 \zeta_{8}^{3} q^{47} + ( 10 + 5 \zeta_{8} - 10 \zeta_{8}^{2} ) q^{48} + 49 q^{49} + ( -24 \zeta_{8} + 7 \zeta_{8}^{3} ) q^{50} + ( -26 + 52 \zeta_{8} + 52 \zeta_{8}^{3} ) q^{51} + ( 24 + 24 \zeta_{8}^{2} ) q^{52} -100 \zeta_{8}^{3} q^{53} + ( -23 + 10 \zeta_{8} + 10 \zeta_{8}^{3} ) q^{54} + ( -49 + 7 \zeta_{8}^{2} ) q^{55} + 49 \zeta_{8}^{3} q^{56} + ( -20 - 10 \zeta_{8} + 20 \zeta_{8}^{2} ) q^{57} + ( -3 - 3 \zeta_{8}^{2} ) q^{58} + ( -21 \zeta_{8} - 21 \zeta_{8}^{3} ) q^{59} + ( 12 - 6 \zeta_{8} + 9 \zeta_{8}^{2} - 42 \zeta_{8}^{3} ) q^{60} + 14 \zeta_{8}^{2} q^{61} -14 \zeta_{8}^{3} q^{62} + ( -28 \zeta_{8} + 49 \zeta_{8}^{2} + 28 \zeta_{8}^{3} ) q^{63} -13 \zeta_{8}^{2} q^{64} + ( 56 \zeta_{8} - 8 \zeta_{8}^{3} ) q^{65} + ( -7 \zeta_{8} + 28 \zeta_{8}^{2} + 7 \zeta_{8}^{3} ) q^{66} + ( 32 - 32 \zeta_{8}^{2} ) q^{67} -78 \zeta_{8} q^{68} + ( 34 - 68 \zeta_{8} - 68 \zeta_{8}^{3} ) q^{69} + ( -21 + 28 \zeta_{8}^{2} ) q^{70} + ( -42 \zeta_{8} - 42 \zeta_{8}^{3} ) q^{71} + ( 28 - 49 \zeta_{8} - 28 \zeta_{8}^{2} ) q^{72} + ( -39 + 39 \zeta_{8}^{2} ) q^{73} + ( 30 \zeta_{8} - 30 \zeta_{8}^{3} ) q^{74} + ( 34 + 24 \zeta_{8} - 62 \zeta_{8}^{2} - 7 \zeta_{8}^{3} ) q^{75} -30 \zeta_{8}^{2} q^{76} + ( 49 \zeta_{8} + 49 \zeta_{8}^{3} ) q^{77} + ( 8 + 8 \zeta_{8}^{2} - 32 \zeta_{8}^{3} ) q^{78} + 56 \zeta_{8}^{2} q^{79} + ( 20 \zeta_{8} + 15 \zeta_{8}^{3} ) q^{80} + ( -17 - 56 \zeta_{8} - 56 \zeta_{8}^{3} ) q^{81} + ( 24 + 24 \zeta_{8}^{2} ) q^{82} + 52 \zeta_{8} q^{83} + ( -42 - 42 \zeta_{8}^{2} - 21 \zeta_{8}^{3} ) q^{84} + ( -104 - 78 \zeta_{8}^{2} ) q^{85} + ( 36 \zeta_{8} + 36 \zeta_{8}^{3} ) q^{86} + ( 3 + 3 \zeta_{8}^{2} - 12 \zeta_{8}^{3} ) q^{87} + ( -49 - 49 \zeta_{8}^{2} ) q^{88} + ( -14 \zeta_{8} - 14 \zeta_{8}^{3} ) q^{89} + ( -28 - 4 \zeta_{8} - 21 \zeta_{8}^{2} - 28 \zeta_{8}^{3} ) q^{90} + ( 56 - 56 \zeta_{8}^{2} ) q^{91} + 102 \zeta_{8} q^{92} + ( 28 + 28 \zeta_{8}^{2} + 14 \zeta_{8}^{3} ) q^{93} + 6 q^{94} + ( -40 \zeta_{8} - 30 \zeta_{8}^{3} ) q^{95} + ( 66 \zeta_{8} + 33 \zeta_{8}^{2} - 66 \zeta_{8}^{3} ) q^{96} + ( -113 - 113 \zeta_{8}^{2} ) q^{97} + 49 \zeta_{8} q^{98} + ( -49 \zeta_{8} - 56 \zeta_{8}^{2} + 49 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 8q^{3} - 28q^{7} + O(q^{10})$$ $$4q - 8q^{3} - 28q^{7} + 12q^{10} + 24q^{12} - 32q^{13} - 12q^{15} - 20q^{16} + 16q^{18} + 40q^{19} + 56q^{21} + 28q^{22} - 28q^{24} - 96q^{25} + 40q^{27} + 8q^{30} - 28q^{33} + 104q^{34} - 84q^{36} + 120q^{37} - 112q^{40} + 144q^{43} - 16q^{45} - 136q^{46} + 40q^{48} + 196q^{49} - 104q^{51} + 96q^{52} - 92q^{54} - 196q^{55} - 80q^{57} - 12q^{58} + 48q^{60} + 128q^{67} + 136q^{69} - 84q^{70} + 112q^{72} - 156q^{73} + 136q^{75} + 32q^{78} - 68q^{81} + 96q^{82} - 168q^{84} - 416q^{85} + 12q^{87} - 196q^{88} - 112q^{90} + 224q^{91} + 112q^{93} + 24q^{94} - 452q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$22$$ $$31$$ $$71$$ $$\chi(n)$$ $$-\zeta_{8}^{2}$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
62.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
−0.707107 + 0.707107i −1.29289 2.70711i 3.00000i 0.707107 4.94975i 2.82843 + 1.00000i −7.00000 −4.94975 4.94975i −5.65685 + 7.00000i 3.00000 + 4.00000i
62.2 0.707107 0.707107i −2.70711 1.29289i 3.00000i −0.707107 + 4.94975i −2.82843 + 1.00000i −7.00000 4.94975 + 4.94975i 5.65685 + 7.00000i 3.00000 + 4.00000i
83.1 −0.707107 0.707107i −1.29289 + 2.70711i 3.00000i 0.707107 + 4.94975i 2.82843 1.00000i −7.00000 −4.94975 + 4.94975i −5.65685 7.00000i 3.00000 4.00000i
83.2 0.707107 + 0.707107i −2.70711 + 1.29289i 3.00000i −0.707107 4.94975i −2.82843 1.00000i −7.00000 4.94975 4.94975i 5.65685 7.00000i 3.00000 4.00000i
 $$n$$: 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
35.f even 4 1 inner
105.k odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.3.k.a 4
3.b odd 2 1 inner 105.3.k.a 4
5.c odd 4 1 105.3.k.b yes 4
7.b odd 2 1 105.3.k.b yes 4
15.e even 4 1 105.3.k.b yes 4
21.c even 2 1 105.3.k.b yes 4
35.f even 4 1 inner 105.3.k.a 4
105.k odd 4 1 inner 105.3.k.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.3.k.a 4 1.a even 1 1 trivial
105.3.k.a 4 3.b odd 2 1 inner
105.3.k.a 4 35.f even 4 1 inner
105.3.k.a 4 105.k odd 4 1 inner
105.3.k.b yes 4 5.c odd 4 1
105.3.k.b yes 4 7.b odd 2 1
105.3.k.b yes 4 15.e even 4 1
105.3.k.b yes 4 21.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 $$S_{3}^{\mathrm{new}}(105, [\chi])$$:

 $$T_{2}^{4} + 1$$ $$T_{11}^{2} + 98$$ $$T_{13}^{2} + 16 T_{13} + 128$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 + 17 T^{4} + 256 T^{8}$$
$3$ $$1 + 8 T + 32 T^{2} + 72 T^{3} + 81 T^{4}$$
$5$ $$1 + 48 T^{2} + 625 T^{4}$$
$7$ $$( 1 + 7 T )^{4}$$
$11$ $$( 1 - 144 T^{2} + 14641 T^{4} )^{2}$$
$13$ $$( 1 + 16 T + 128 T^{2} + 2704 T^{3} + 28561 T^{4} )^{2}$$
$17$ $$1 - 157438 T^{4} + 6975757441 T^{8}$$
$19$ $$( 1 - 10 T + 361 T^{2} )^{4}$$
$23$ $$1 - 550078 T^{4} + 78310985281 T^{8}$$
$29$ $$( 1 + 1664 T^{2} + 707281 T^{4} )^{2}$$
$31$ $$( 1 - 1726 T^{2} + 923521 T^{4} )^{2}$$
$37$ $$( 1 - 60 T + 1800 T^{2} - 82140 T^{3} + 1874161 T^{4} )^{2}$$
$41$ $$( 1 + 2210 T^{2} + 2825761 T^{4} )^{2}$$
$43$ $$( 1 - 72 T + 2592 T^{2} - 133128 T^{3} + 3418801 T^{4} )^{2}$$
$47$ $$1 + 9442562 T^{4} + 23811286661761 T^{8}$$
$53$ $$1 + 3420962 T^{4} + 62259690411361 T^{8}$$
$59$ $$( 1 - 6080 T^{2} + 12117361 T^{4} )^{2}$$
$61$ $$( 1 - 7246 T^{2} + 13845841 T^{4} )^{2}$$
$67$ $$( 1 - 64 T + 2048 T^{2} - 287296 T^{3} + 20151121 T^{4} )^{2}$$
$71$ $$( 1 - 6554 T^{2} + 25411681 T^{4} )^{2}$$
$73$ $$( 1 + 78 T + 3042 T^{2} + 415662 T^{3} + 28398241 T^{4} )^{2}$$
$79$ $$( 1 - 9346 T^{2} + 38950081 T^{4} )^{2}$$
$83$ $$1 + 27716834 T^{4} + 2252292232139041 T^{8}$$
$89$ $$( 1 - 15450 T^{2} + 62742241 T^{4} )^{2}$$
$97$ $$( 1 + 226 T + 25538 T^{2} + 2126434 T^{3} + 88529281 T^{4} )^{2}$$