# Properties

 Label 30.6.c.a Level $30$ Weight $6$ Character orbit 30.c Analytic conductor $4.812$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$30 = 2 \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$6$$ Character orbit: $$[\chi]$$ $$=$$ 30.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.81151459439$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + 4 i q^{2} + 9 i q^{3} -16 q^{4} + ( -55 + 10 i ) q^{5} -36 q^{6} -4 i q^{7} -64 i q^{8} -81 q^{9} +O(q^{10})$$ $$q + 4 i q^{2} + 9 i q^{3} -16 q^{4} + ( -55 + 10 i ) q^{5} -36 q^{6} -4 i q^{7} -64 i q^{8} -81 q^{9} + ( -40 - 220 i ) q^{10} -500 q^{11} -144 i q^{12} + 288 i q^{13} + 16 q^{14} + ( -90 - 495 i ) q^{15} + 256 q^{16} + 1516 i q^{17} -324 i q^{18} + 1344 q^{19} + ( 880 - 160 i ) q^{20} + 36 q^{21} -2000 i q^{22} + 4100 i q^{23} + 576 q^{24} + ( 2925 - 1100 i ) q^{25} -1152 q^{26} -729 i q^{27} + 64 i q^{28} + 2646 q^{29} + ( 1980 - 360 i ) q^{30} -5612 q^{31} + 1024 i q^{32} -4500 i q^{33} -6064 q^{34} + ( 40 + 220 i ) q^{35} + 1296 q^{36} -7288 i q^{37} + 5376 i q^{38} -2592 q^{39} + ( 640 + 3520 i ) q^{40} -18986 q^{41} + 144 i q^{42} + 2404 i q^{43} + 8000 q^{44} + ( 4455 - 810 i ) q^{45} -16400 q^{46} + 8900 i q^{47} + 2304 i q^{48} + 16791 q^{49} + ( 4400 + 11700 i ) q^{50} -13644 q^{51} -4608 i q^{52} -39804 i q^{53} + 2916 q^{54} + ( 27500 - 5000 i ) q^{55} -256 q^{56} + 12096 i q^{57} + 10584 i q^{58} + 28300 q^{59} + ( 1440 + 7920 i ) q^{60} + 18290 q^{61} -22448 i q^{62} + 324 i q^{63} -4096 q^{64} + ( -2880 - 15840 i ) q^{65} + 18000 q^{66} + 65956 i q^{67} -24256 i q^{68} -36900 q^{69} + ( -880 + 160 i ) q^{70} -28800 q^{71} + 5184 i q^{72} + 30808 i q^{73} + 29152 q^{74} + ( 9900 + 26325 i ) q^{75} -21504 q^{76} + 2000 i q^{77} -10368 i q^{78} -60228 q^{79} + ( -14080 + 2560 i ) q^{80} + 6561 q^{81} -75944 i q^{82} + 2468 i q^{83} -576 q^{84} + ( -15160 - 83380 i ) q^{85} -9616 q^{86} + 23814 i q^{87} + 32000 i q^{88} -22678 q^{89} + ( 3240 + 17820 i ) q^{90} + 1152 q^{91} -65600 i q^{92} -50508 i q^{93} -35600 q^{94} + ( -73920 + 13440 i ) q^{95} -9216 q^{96} -36968 i q^{97} + 67164 i q^{98} + 40500 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 32q^{4} - 110q^{5} - 72q^{6} - 162q^{9} + O(q^{10})$$ $$2q - 32q^{4} - 110q^{5} - 72q^{6} - 162q^{9} - 80q^{10} - 1000q^{11} + 32q^{14} - 180q^{15} + 512q^{16} + 2688q^{19} + 1760q^{20} + 72q^{21} + 1152q^{24} + 5850q^{25} - 2304q^{26} + 5292q^{29} + 3960q^{30} - 11224q^{31} - 12128q^{34} + 80q^{35} + 2592q^{36} - 5184q^{39} + 1280q^{40} - 37972q^{41} + 16000q^{44} + 8910q^{45} - 32800q^{46} + 33582q^{49} + 8800q^{50} - 27288q^{51} + 5832q^{54} + 55000q^{55} - 512q^{56} + 56600q^{59} + 2880q^{60} + 36580q^{61} - 8192q^{64} - 5760q^{65} + 36000q^{66} - 73800q^{69} - 1760q^{70} - 57600q^{71} + 58304q^{74} + 19800q^{75} - 43008q^{76} - 120456q^{79} - 28160q^{80} + 13122q^{81} - 1152q^{84} - 30320q^{85} - 19232q^{86} - 45356q^{89} + 6480q^{90} + 2304q^{91} - 71200q^{94} - 147840q^{95} - 18432q^{96} + 81000q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/30\mathbb{Z}\right)^\times$$.

 $$n$$ $$7$$ $$11$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1
 − 1.00000i 1.00000i
4.00000i 9.00000i −16.0000 −55.0000 10.0000i −36.0000 4.00000i 64.0000i −81.0000 −40.0000 + 220.000i
19.2 4.00000i 9.00000i −16.0000 −55.0000 + 10.0000i −36.0000 4.00000i 64.0000i −81.0000 −40.0000 220.000i
 $$n$$: 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 30.6.c.a 2
3.b odd 2 1 90.6.c.b 2
4.b odd 2 1 240.6.f.a 2
5.b even 2 1 inner 30.6.c.a 2
5.c odd 4 1 150.6.a.f 1
5.c odd 4 1 150.6.a.j 1
12.b even 2 1 720.6.f.g 2
15.d odd 2 1 90.6.c.b 2
15.e even 4 1 450.6.a.f 1
15.e even 4 1 450.6.a.s 1
20.d odd 2 1 240.6.f.a 2
60.h even 2 1 720.6.f.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.c.a 2 1.a even 1 1 trivial
30.6.c.a 2 5.b even 2 1 inner
90.6.c.b 2 3.b odd 2 1
90.6.c.b 2 15.d odd 2 1
150.6.a.f 1 5.c odd 4 1
150.6.a.j 1 5.c odd 4 1
240.6.f.a 2 4.b odd 2 1
240.6.f.a 2 20.d odd 2 1
450.6.a.f 1 15.e even 4 1
450.6.a.s 1 15.e even 4 1
720.6.f.g 2 12.b even 2 1
720.6.f.g 2 60.h even 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{7}^{2} + 16$$ acting on $$S_{6}^{\mathrm{new}}(30, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$16 + T^{2}$$
$3$ $$81 + T^{2}$$
$5$ $$3125 + 110 T + T^{2}$$
$7$ $$16 + T^{2}$$
$11$ $$( 500 + T )^{2}$$
$13$ $$82944 + T^{2}$$
$17$ $$2298256 + T^{2}$$
$19$ $$( -1344 + T )^{2}$$
$23$ $$16810000 + T^{2}$$
$29$ $$( -2646 + T )^{2}$$
$31$ $$( 5612 + T )^{2}$$
$37$ $$53114944 + T^{2}$$
$41$ $$( 18986 + T )^{2}$$
$43$ $$5779216 + T^{2}$$
$47$ $$79210000 + T^{2}$$
$53$ $$1584358416 + T^{2}$$
$59$ $$( -28300 + T )^{2}$$
$61$ $$( -18290 + T )^{2}$$
$67$ $$4350193936 + T^{2}$$
$71$ $$( 28800 + T )^{2}$$
$73$ $$949132864 + T^{2}$$
$79$ $$( 60228 + T )^{2}$$
$83$ $$6091024 + T^{2}$$
$89$ $$( 22678 + T )^{2}$$
$97$ $$1366633024 + T^{2}$$