# Properties

 Label 105.4.d.a Level $105$ Weight $4$ Character orbit 105.d Analytic conductor $6.195$ Analytic rank $0$ Dimension $6$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$105 = 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 105.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.19520055060$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.84052224.1 Defining polynomial: $$x^{6} - 2 x^{5} + 2 x^{4} + 6 x^{3} + 36 x^{2} - 36 x + 18$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2^{2}$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -3 \beta_{3} q^{3} + ( -1 - 2 \beta_{2} - \beta_{4} + \beta_{5} ) q^{4} + ( -3 + 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{5} -3 \beta_{2} q^{6} + 7 \beta_{3} q^{7} + ( \beta_{1} + 8 \beta_{3} + \beta_{4} + \beta_{5} ) q^{8} -9 q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -3 \beta_{3} q^{3} + ( -1 - 2 \beta_{2} - \beta_{4} + \beta_{5} ) q^{4} + ( -3 + 2 \beta_{1} - \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{5} -3 \beta_{2} q^{6} + 7 \beta_{3} q^{7} + ( \beta_{1} + 8 \beta_{3} + \beta_{4} + \beta_{5} ) q^{8} -9 q^{9} + ( -13 - 8 \beta_{1} - 6 \beta_{2} - 6 \beta_{3} - 2 \beta_{4} + \beta_{5} ) q^{10} + ( -26 + 4 \beta_{2} - 4 \beta_{4} + 4 \beta_{5} ) q^{11} + ( -6 \beta_{1} + 3 \beta_{3} + 3 \beta_{4} + 3 \beta_{5} ) q^{12} + ( 4 \beta_{1} + 36 \beta_{3} + \beta_{4} + \beta_{5} ) q^{13} + 7 \beta_{2} q^{14} + ( -3 - 3 \beta_{1} - 6 \beta_{2} + 9 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} ) q^{15} + ( -27 - 8 \beta_{2} - 10 \beta_{4} + 10 \beta_{5} ) q^{16} + ( -8 \beta_{1} - 50 \beta_{3} + 5 \beta_{4} + 5 \beta_{5} ) q^{17} -9 \beta_{1} q^{18} + ( 44 + 8 \beta_{2} + \beta_{4} - \beta_{5} ) q^{19} + ( 53 - 12 \beta_{1} + \beta_{2} + 31 \beta_{3} - 3 \beta_{4} + 4 \beta_{5} ) q^{20} + 21 q^{21} + ( -26 \beta_{1} - 76 \beta_{3} - 8 \beta_{4} - 8 \beta_{5} ) q^{22} + ( 22 \beta_{1} - 52 \beta_{3} + \beta_{4} + \beta_{5} ) q^{23} + ( 24 - 3 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} ) q^{24} + ( 67 - 18 \beta_{1} - 26 \beta_{2} + 44 \beta_{3} - 7 \beta_{4} + \beta_{5} ) q^{25} + ( -46 + 30 \beta_{2} - 5 \beta_{4} + 5 \beta_{5} ) q^{26} + 27 \beta_{3} q^{27} + ( 14 \beta_{1} - 7 \beta_{3} - 7 \beta_{4} - 7 \beta_{5} ) q^{28} + ( -66 + 40 \beta_{2} + 6 \beta_{4} - 6 \beta_{5} ) q^{29} + ( -18 - 18 \beta_{1} + 24 \beta_{2} + 39 \beta_{3} + 3 \beta_{4} + 6 \beta_{5} ) q^{30} + ( -104 - 40 \beta_{2} + 7 \beta_{4} - 7 \beta_{5} ) q^{31} + ( -55 \beta_{1} + 36 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} ) q^{32} + ( 12 \beta_{1} + 78 \beta_{3} + 12 \beta_{4} + 12 \beta_{5} ) q^{33} + ( 22 - 24 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} ) q^{34} + ( 7 + 7 \beta_{1} + 14 \beta_{2} - 21 \beta_{3} - 7 \beta_{4} - 14 \beta_{5} ) q^{35} + ( 9 + 18 \beta_{2} + 9 \beta_{4} - 9 \beta_{5} ) q^{36} + ( 64 \beta_{1} - 84 \beta_{3} - 32 \beta_{4} - 32 \beta_{5} ) q^{37} + ( 62 \beta_{1} - 62 \beta_{3} - 7 \beta_{4} - 7 \beta_{5} ) q^{38} + ( 108 - 12 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} ) q^{39} + ( -1 - 16 \beta_{1} + 8 \beta_{2} - 92 \beta_{3} - 9 \beta_{4} - 8 \beta_{5} ) q^{40} + ( -34 - 18 \beta_{2} + 43 \beta_{4} - 43 \beta_{5} ) q^{41} + 21 \beta_{1} q^{42} + ( 120 \beta_{1} - 4 \beta_{3} - 12 \beta_{4} - 12 \beta_{5} ) q^{43} + ( 106 - 8 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} ) q^{44} + ( 27 - 18 \beta_{1} + 9 \beta_{2} + 9 \beta_{3} + 18 \beta_{4} - 9 \beta_{5} ) q^{45} + ( -208 - 94 \beta_{2} - 23 \beta_{4} + 23 \beta_{5} ) q^{46} + ( 32 \beta_{1} + 52 \beta_{3} + 50 \beta_{4} + 50 \beta_{5} ) q^{47} + ( -24 \beta_{1} + 81 \beta_{3} + 30 \beta_{4} + 30 \beta_{5} ) q^{48} -49 q^{49} + ( 192 + 7 \beta_{1} + 74 \beta_{2} + 194 \beta_{3} + 43 \beta_{4} + \beta_{5} ) q^{50} + ( -150 + 24 \beta_{2} + 15 \beta_{4} - 15 \beta_{5} ) q^{51} + ( 36 \beta_{1} - 32 \beta_{3} - 27 \beta_{4} - 27 \beta_{5} ) q^{52} + ( 132 \beta_{1} + 144 \beta_{3} - 27 \beta_{4} - 27 \beta_{5} ) q^{53} + 27 \beta_{2} q^{54} + ( 350 - 20 \beta_{1} - 70 \beta_{2} - 10 \beta_{3} + 20 \beta_{4} - 30 \beta_{5} ) q^{55} + ( -56 + 7 \beta_{2} - 7 \beta_{4} + 7 \beta_{5} ) q^{56} + ( 24 \beta_{1} - 132 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} ) q^{57} + ( 26 \beta_{1} - 300 \beta_{3} - 34 \beta_{4} - 34 \beta_{5} ) q^{58} + ( 332 - 108 \beta_{2} + 4 \beta_{4} - 4 \beta_{5} ) q^{59} + ( 93 + 3 \beta_{1} + 36 \beta_{2} - 159 \beta_{3} + 12 \beta_{4} + 9 \beta_{5} ) q^{60} + ( -306 + 164 \beta_{2} + 58 \beta_{4} - 58 \beta_{5} ) q^{61} + ( -170 \beta_{1} + 430 \beta_{3} + 47 \beta_{4} + 47 \beta_{5} ) q^{62} -63 \beta_{3} q^{63} + ( 219 + 94 \beta_{2} - 31 \beta_{4} + 31 \beta_{5} ) q^{64} + ( -12 - 12 \beta_{1} + 46 \beta_{2} - 194 \beta_{3} - 43 \beta_{4} - 61 \beta_{5} ) q^{65} + ( -228 + 78 \beta_{2} - 24 \beta_{4} + 24 \beta_{5} ) q^{66} + ( -24 \beta_{1} - 20 \beta_{3} - 66 \beta_{4} - 66 \beta_{5} ) q^{67} + ( -84 \beta_{1} - 154 \beta_{3} + 67 \beta_{4} + 67 \beta_{5} ) q^{68} + ( -156 - 66 \beta_{2} + 3 \beta_{4} - 3 \beta_{5} ) q^{69} + ( 42 + 42 \beta_{1} - 56 \beta_{2} - 91 \beta_{3} - 7 \beta_{4} - 14 \beta_{5} ) q^{70} + ( -374 + 168 \beta_{2} + 32 \beta_{4} - 32 \beta_{5} ) q^{71} + ( -9 \beta_{1} - 72 \beta_{3} - 9 \beta_{4} - 9 \beta_{5} ) q^{72} + ( -68 \beta_{1} + 452 \beta_{3} - 65 \beta_{4} - 65 \beta_{5} ) q^{73} + ( -256 - 276 \beta_{2} - 32 \beta_{4} + 32 \beta_{5} ) q^{74} + ( 132 - 78 \beta_{1} + 54 \beta_{2} - 201 \beta_{3} + 3 \beta_{4} + 21 \beta_{5} ) q^{75} + ( -136 - 136 \beta_{2} - 47 \beta_{4} + 47 \beta_{5} ) q^{76} + ( -28 \beta_{1} - 182 \beta_{3} - 28 \beta_{4} - 28 \beta_{5} ) q^{77} + ( 90 \beta_{1} + 138 \beta_{3} + 15 \beta_{4} + 15 \beta_{5} ) q^{78} + ( 464 + 168 \beta_{2} - 30 \beta_{4} + 30 \beta_{5} ) q^{79} + ( 653 - 82 \beta_{1} - 69 \beta_{2} + 171 \beta_{3} - 8 \beta_{4} - \beta_{5} ) q^{80} + 81 q^{81} + ( 16 \beta_{1} + 592 \beta_{3} + 61 \beta_{4} + 61 \beta_{5} ) q^{82} + ( -308 \beta_{1} - 356 \beta_{3} + 56 \beta_{4} + 56 \beta_{5} ) q^{83} + ( -21 - 42 \beta_{2} - 21 \beta_{4} + 21 \beta_{5} ) q^{84} + ( 74 - 66 \beta_{1} - 62 \beta_{2} - 112 \beta_{3} + 71 \beta_{4} + 127 \beta_{5} ) q^{85} + ( -960 - 268 \beta_{2} - 108 \beta_{4} + 108 \beta_{5} ) q^{86} + ( 120 \beta_{1} + 198 \beta_{3} - 18 \beta_{4} - 18 \beta_{5} ) q^{87} + ( -114 \beta_{1} - 516 \beta_{3} - 54 \beta_{4} - 54 \beta_{5} ) q^{88} + ( 650 + 74 \beta_{2} + 43 \beta_{4} - 43 \beta_{5} ) q^{89} + ( 117 + 72 \beta_{1} + 54 \beta_{2} + 54 \beta_{3} + 18 \beta_{4} - 9 \beta_{5} ) q^{90} + ( -252 + 28 \beta_{2} - 7 \beta_{4} + 7 \beta_{5} ) q^{91} + ( -266 \beta_{1} + 200 \beta_{3} + 79 \beta_{4} + 79 \beta_{5} ) q^{92} + ( -120 \beta_{1} + 312 \beta_{3} - 21 \beta_{4} - 21 \beta_{5} ) q^{93} + ( -788 + 88 \beta_{2} - 82 \beta_{4} + 82 \beta_{5} ) q^{94} + ( -146 + 134 \beta_{1} - 92 \beta_{2} - 152 \beta_{3} - 89 \beta_{4} + 27 \beta_{5} ) q^{95} + ( 108 + 165 \beta_{2} + 18 \beta_{4} - 18 \beta_{5} ) q^{96} + ( 156 \beta_{1} - 556 \beta_{3} + 27 \beta_{4} + 27 \beta_{5} ) q^{97} -49 \beta_{1} q^{98} + ( 234 - 36 \beta_{2} + 36 \beta_{4} - 36 \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 6q^{4} - 14q^{5} - 6q^{6} - 54q^{9} + O(q^{10})$$ $$6q - 6q^{4} - 14q^{5} - 6q^{6} - 54q^{9} - 84q^{10} - 132q^{11} + 14q^{14} - 24q^{15} - 138q^{16} + 276q^{19} + 334q^{20} + 126q^{21} + 126q^{24} + 366q^{25} - 196q^{26} - 340q^{29} - 54q^{30} - 732q^{31} + 72q^{34} + 56q^{35} + 54q^{36} + 612q^{39} + 12q^{40} - 412q^{41} + 612q^{44} + 126q^{45} - 1344q^{46} - 294q^{49} + 1216q^{50} - 912q^{51} + 54q^{54} + 1860q^{55} - 294q^{56} + 1760q^{59} + 624q^{60} - 1740q^{61} + 1626q^{64} - 16q^{65} - 1116q^{66} - 1080q^{69} + 126q^{70} - 2036q^{71} - 1960q^{74} + 936q^{75} - 900q^{76} + 3240q^{79} + 3794q^{80} + 486q^{81} - 126q^{84} + 432q^{85} - 5864q^{86} + 3876q^{89} + 756q^{90} - 1428q^{91} - 4224q^{94} - 828q^{95} + 906q^{96} + 1188q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} + 2 x^{4} + 6 x^{3} + 36 x^{2} - 36 x + 18$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-7 \nu^{5} + 3 \nu^{4} + 53 \nu^{3} - 130 \nu^{2} + 42 \nu - 9$$$$)/327$$ $$\beta_{2}$$ $$=$$ $$($$$$8 \nu^{5} - 19 \nu^{4} + 64 \nu^{3} + 24 \nu^{2} - 48 \nu + 57$$$$)/327$$ $$\beta_{3}$$ $$=$$ $$($$$$15 \nu^{5} - 22 \nu^{4} + 11 \nu^{3} + 154 \nu^{2} + 564 \nu - 261$$$$)/327$$ $$\beta_{4}$$ $$=$$ $$($$$$-72 \nu^{5} + 62 \nu^{4} - 31 \nu^{3} - 434 \nu^{2} - 2838 \nu - 186$$$$)/327$$ $$\beta_{5}$$ $$=$$ $$($$$$-78 \nu^{5} + 158 \nu^{4} - 79 \nu^{3} - 452 \nu^{2} - 2802 \nu + 2796$$$$)/327$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{5} + \beta_{4} + 10 \beta_{3}$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$\beta_{5} + 5 \beta_{3} + 3 \beta_{2} + 3 \beta_{1} - 5$$ $$\nu^{4}$$ $$=$$ $$4 \beta_{5} - 4 \beta_{4} + 3 \beta_{2} - 37$$ $$\nu^{5}$$ $$=$$ $$-11 \beta_{4} - 52 \beta_{3} + 21 \beta_{2} - 21 \beta_{1} - 52$$

## Character values

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

 $$n$$ $$22$$ $$31$$ $$71$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
64.1
 −1.55322 − 1.55322i 2.09148 − 2.09148i 0.461746 + 0.461746i 0.461746 − 0.461746i 2.09148 + 2.09148i −1.55322 + 1.55322i
4.10645i 3.00000i −8.86293 −6.58150 9.03792i −12.3193 7.00000i 3.54358i −9.00000 −37.1137 + 27.0266i
64.2 3.18296i 3.00000i −2.13122 −11.0627 1.61735i 9.54887 7.00000i 18.6801i −9.00000 −5.14795 + 35.2122i
64.3 0.0765073i 3.00000i 7.99415 10.6442 + 3.42057i −0.229522 7.00000i 1.22367i −9.00000 0.261698 0.814361i
64.4 0.0765073i 3.00000i 7.99415 10.6442 3.42057i −0.229522 7.00000i 1.22367i −9.00000 0.261698 + 0.814361i
64.5 3.18296i 3.00000i −2.13122 −11.0627 + 1.61735i 9.54887 7.00000i 18.6801i −9.00000 −5.14795 35.2122i
64.6 4.10645i 3.00000i −8.86293 −6.58150 + 9.03792i −12.3193 7.00000i 3.54358i −9.00000 −37.1137 27.0266i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 64.6 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 105.4.d.a 6
3.b odd 2 1 315.4.d.a 6
5.b even 2 1 inner 105.4.d.a 6
5.c odd 4 1 525.4.a.q 3
5.c odd 4 1 525.4.a.r 3
15.d odd 2 1 315.4.d.a 6
15.e even 4 1 1575.4.a.bc 3
15.e even 4 1 1575.4.a.bd 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.4.d.a 6 1.a even 1 1 trivial
105.4.d.a 6 5.b even 2 1 inner
315.4.d.a 6 3.b odd 2 1
315.4.d.a 6 15.d odd 2 1
525.4.a.q 3 5.c odd 4 1
525.4.a.r 3 5.c odd 4 1
1575.4.a.bc 3 15.e even 4 1
1575.4.a.bd 3 15.e even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} + 27 T_{2}^{4} + 171 T_{2}^{2} + 1$$ acting on $$S_{4}^{\mathrm{new}}(105, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 171 T^{2} + 27 T^{4} + T^{6}$$
$3$ $$( 9 + T^{2} )^{3}$$
$5$ $$1953125 + 218750 T - 10625 T^{2} - 2700 T^{3} - 85 T^{4} + 14 T^{5} + T^{6}$$
$7$ $$( 49 + T^{2} )^{3}$$
$11$ $$( -6120 - 276 T + 66 T^{2} + T^{3} )^{2}$$
$13$ $$823460416 + 3657072 T^{2} + 4236 T^{4} + T^{6}$$
$17$ $$4661065984 + 16797696 T^{2} + 10848 T^{4} + T^{6}$$
$19$ $$( -70632 + 5628 T - 138 T^{2} + T^{3} )^{2}$$
$23$ $$52356761856 + 96212160 T^{2} + 24960 T^{4} + T^{6}$$
$29$ $$( -1680584 - 8116 T + 170 T^{2} + T^{3} )^{2}$$
$31$ $$( -3510632 + 12828 T + 366 T^{2} + T^{3} )^{2}$$
$37$ $$67599442284544 + 6906385152 T^{2} + 168624 T^{4} + T^{6}$$
$41$ $$( -18222200 - 137980 T + 206 T^{2} + T^{3} )^{2}$$
$43$ $$187722443001856 + 23160275712 T^{2} + 334896 T^{4} + T^{6}$$
$47$ $$436551881342976 + 50675116032 T^{2} + 455232 T^{4} + T^{6}$$
$53$ $$2866407388545600 + 60647345136 T^{2} + 426924 T^{4} + T^{6}$$
$59$ $$( 22878976 + 90560 T - 880 T^{2} + T^{3} )^{2}$$
$61$ $$( -112468792 - 98580 T + 870 T^{2} + T^{3} )^{2}$$
$67$ $$1238328217452544 + 126993408000 T^{2} + 702720 T^{4} + T^{6}$$
$71$ $$( -133243912 + 34316 T + 1018 T^{2} + T^{3} )^{2}$$
$73$ $$39858579619730496 + 735022872432 T^{2} + 1738092 T^{4} + T^{6}$$
$79$ $$( 258624448 + 308400 T - 1620 T^{2} + T^{3} )^{2}$$
$83$ $$473853851651866624 + 1827579297792 T^{2} + 2344128 T^{4} + T^{6}$$
$89$ $$( -181473048 + 1131348 T - 1938 T^{2} + T^{3} )^{2}$$
$97$ $$19742608222425664 + 1009246730352 T^{2} + 2139084 T^{4} + T^{6}$$