# Properties

 Label 105.2.d.a Level 105 Weight 2 Character orbit 105.d Analytic conductor 0.838 Analytic rank 0 Dimension 2 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.838429221223$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} + i q^{3} + q^{4} + ( 1 - 2 i ) q^{5} - q^{6} + i q^{7} + 3 i q^{8} - q^{9} +O(q^{10})$$ $$q + i q^{2} + i q^{3} + q^{4} + ( 1 - 2 i ) q^{5} - q^{6} + i q^{7} + 3 i q^{8} - q^{9} + ( 2 + i ) q^{10} -6 q^{11} + i q^{12} -2 i q^{13} - q^{14} + ( 2 + i ) q^{15} - q^{16} -4 i q^{17} -i q^{18} + 6 q^{19} + ( 1 - 2 i ) q^{20} - q^{21} -6 i q^{22} -3 q^{24} + ( -3 - 4 i ) q^{25} + 2 q^{26} -i q^{27} + i q^{28} + 2 q^{29} + ( -1 + 2 i ) q^{30} -10 q^{31} + 5 i q^{32} -6 i q^{33} + 4 q^{34} + ( 2 + i ) q^{35} - q^{36} + 4 i q^{37} + 6 i q^{38} + 2 q^{39} + ( 6 + 3 i ) q^{40} + 2 q^{41} -i q^{42} -4 i q^{43} -6 q^{44} + ( -1 + 2 i ) q^{45} -i q^{48} - q^{49} + ( 4 - 3 i ) q^{50} + 4 q^{51} -2 i q^{52} + 6 i q^{53} + q^{54} + ( -6 + 12 i ) q^{55} -3 q^{56} + 6 i q^{57} + 2 i q^{58} + 8 q^{59} + ( 2 + i ) q^{60} -2 q^{61} -10 i q^{62} -i q^{63} -7 q^{64} + ( -4 - 2 i ) q^{65} + 6 q^{66} + 16 i q^{67} -4 i q^{68} + ( -1 + 2 i ) q^{70} + 10 q^{71} -3 i q^{72} -6 i q^{73} -4 q^{74} + ( 4 - 3 i ) q^{75} + 6 q^{76} -6 i q^{77} + 2 i q^{78} -4 q^{79} + ( -1 + 2 i ) q^{80} + q^{81} + 2 i q^{82} + 8 i q^{83} - q^{84} + ( -8 - 4 i ) q^{85} + 4 q^{86} + 2 i q^{87} -18 i q^{88} -6 q^{89} + ( -2 - i ) q^{90} + 2 q^{91} -10 i q^{93} + ( 6 - 12 i ) q^{95} -5 q^{96} + 2 i q^{97} -i q^{98} + 6 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{4} + 2q^{5} - 2q^{6} - 2q^{9} + O(q^{10})$$ $$2q + 2q^{4} + 2q^{5} - 2q^{6} - 2q^{9} + 4q^{10} - 12q^{11} - 2q^{14} + 4q^{15} - 2q^{16} + 12q^{19} + 2q^{20} - 2q^{21} - 6q^{24} - 6q^{25} + 4q^{26} + 4q^{29} - 2q^{30} - 20q^{31} + 8q^{34} + 4q^{35} - 2q^{36} + 4q^{39} + 12q^{40} + 4q^{41} - 12q^{44} - 2q^{45} - 2q^{49} + 8q^{50} + 8q^{51} + 2q^{54} - 12q^{55} - 6q^{56} + 16q^{59} + 4q^{60} - 4q^{61} - 14q^{64} - 8q^{65} + 12q^{66} - 2q^{70} + 20q^{71} - 8q^{74} + 8q^{75} + 12q^{76} - 8q^{79} - 2q^{80} + 2q^{81} - 2q^{84} - 16q^{85} + 8q^{86} - 12q^{89} - 4q^{90} + 4q^{91} + 12q^{95} - 10q^{96} + 12q^{99} + O(q^{100})$$

## Character values

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

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

## 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
 − 1.00000i 1.00000i
1.00000i 1.00000i 1.00000 1.00000 + 2.00000i −1.00000 1.00000i 3.00000i −1.00000 2.00000 1.00000i
64.2 1.00000i 1.00000i 1.00000 1.00000 2.00000i −1.00000 1.00000i 3.00000i −1.00000 2.00000 + 1.00000i
 $$n$$: 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
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 105.2.d.a 2
3.b odd 2 1 315.2.d.c 2
4.b odd 2 1 1680.2.t.f 2
5.b even 2 1 inner 105.2.d.a 2
5.c odd 4 1 525.2.a.b 1
5.c odd 4 1 525.2.a.c 1
7.b odd 2 1 735.2.d.a 2
7.c even 3 2 735.2.q.a 4
7.d odd 6 2 735.2.q.b 4
12.b even 2 1 5040.2.t.e 2
15.d odd 2 1 315.2.d.c 2
15.e even 4 1 1575.2.a.e 1
15.e even 4 1 1575.2.a.i 1
20.d odd 2 1 1680.2.t.f 2
20.e even 4 1 8400.2.a.bj 1
20.e even 4 1 8400.2.a.ch 1
21.c even 2 1 2205.2.d.f 2
35.c odd 2 1 735.2.d.a 2
35.f even 4 1 3675.2.a.d 1
35.f even 4 1 3675.2.a.l 1
35.i odd 6 2 735.2.q.b 4
35.j even 6 2 735.2.q.a 4
60.h even 2 1 5040.2.t.e 2
105.g even 2 1 2205.2.d.f 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.d.a 2 1.a even 1 1 trivial
105.2.d.a 2 5.b even 2 1 inner
315.2.d.c 2 3.b odd 2 1
315.2.d.c 2 15.d odd 2 1
525.2.a.b 1 5.c odd 4 1
525.2.a.c 1 5.c odd 4 1
735.2.d.a 2 7.b odd 2 1
735.2.d.a 2 35.c odd 2 1
735.2.q.a 4 7.c even 3 2
735.2.q.a 4 35.j even 6 2
735.2.q.b 4 7.d odd 6 2
735.2.q.b 4 35.i odd 6 2
1575.2.a.e 1 15.e even 4 1
1575.2.a.i 1 15.e even 4 1
1680.2.t.f 2 4.b odd 2 1
1680.2.t.f 2 20.d odd 2 1
2205.2.d.f 2 21.c even 2 1
2205.2.d.f 2 105.g even 2 1
3675.2.a.d 1 35.f even 4 1
3675.2.a.l 1 35.f even 4 1
5040.2.t.e 2 12.b even 2 1
5040.2.t.e 2 60.h even 2 1
8400.2.a.bj 1 20.e even 4 1
8400.2.a.ch 1 20.e even 4 1

## Hecke kernels

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

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - 3 T^{2} + 4 T^{4}$$
$3$ $$1 + T^{2}$$
$5$ $$1 - 2 T + 5 T^{2}$$
$7$ $$1 + T^{2}$$
$11$ $$( 1 + 6 T + 11 T^{2} )^{2}$$
$13$ $$1 - 22 T^{2} + 169 T^{4}$$
$17$ $$1 - 18 T^{2} + 289 T^{4}$$
$19$ $$( 1 - 6 T + 19 T^{2} )^{2}$$
$23$ $$( 1 - 23 T^{2} )^{2}$$
$29$ $$( 1 - 2 T + 29 T^{2} )^{2}$$
$31$ $$( 1 + 10 T + 31 T^{2} )^{2}$$
$37$ $$1 - 58 T^{2} + 1369 T^{4}$$
$41$ $$( 1 - 2 T + 41 T^{2} )^{2}$$
$43$ $$1 - 70 T^{2} + 1849 T^{4}$$
$47$ $$( 1 - 47 T^{2} )^{2}$$
$53$ $$1 - 70 T^{2} + 2809 T^{4}$$
$59$ $$( 1 - 8 T + 59 T^{2} )^{2}$$
$61$ $$( 1 + 2 T + 61 T^{2} )^{2}$$
$67$ $$1 + 122 T^{2} + 4489 T^{4}$$
$71$ $$( 1 - 10 T + 71 T^{2} )^{2}$$
$73$ $$( 1 - 16 T + 73 T^{2} )( 1 + 16 T + 73 T^{2} )$$
$79$ $$( 1 + 4 T + 79 T^{2} )^{2}$$
$83$ $$1 - 102 T^{2} + 6889 T^{4}$$
$89$ $$( 1 + 6 T + 89 T^{2} )^{2}$$
$97$ $$1 - 190 T^{2} + 9409 T^{4}$$