# Properties

 Label 30.2.e.a Level 30 Weight 2 Character orbit 30.e Analytic conductor 0.240 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$30 = 2 \cdot 3 \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 30.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.239551206064$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ 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.

 $$f(q)$$ $$=$$ $$q + \zeta_{8} q^{2} + ( -1 - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + \zeta_{8}^{2} q^{4} + ( -2 \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( -1 - \zeta_{8} - \zeta_{8}^{3} ) q^{6} + ( -1 + \zeta_{8}^{2} ) q^{7} + \zeta_{8}^{3} q^{8} + ( 2 \zeta_{8} + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{9} +O(q^{10})$$ $$q + \zeta_{8} q^{2} + ( -1 - \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{3} + \zeta_{8}^{2} q^{4} + ( -2 \zeta_{8} - \zeta_{8}^{3} ) q^{5} + ( -1 - \zeta_{8} - \zeta_{8}^{3} ) q^{6} + ( -1 + \zeta_{8}^{2} ) q^{7} + \zeta_{8}^{3} q^{8} + ( 2 \zeta_{8} + \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{9} + ( 1 - 2 \zeta_{8}^{2} ) q^{10} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{11} + ( 1 - \zeta_{8} - \zeta_{8}^{2} ) q^{12} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{14} + ( 2 + \zeta_{8} + \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{15} - q^{16} + 2 \zeta_{8} q^{17} + ( 2 + 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{18} -4 \zeta_{8}^{2} q^{19} + ( \zeta_{8} - 2 \zeta_{8}^{3} ) q^{20} + ( 2 - \zeta_{8} - \zeta_{8}^{3} ) q^{21} + ( -1 + \zeta_{8}^{2} ) q^{22} -4 \zeta_{8}^{3} q^{23} + ( \zeta_{8} - \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{24} + ( -4 + 3 \zeta_{8}^{2} ) q^{25} + ( -1 - 5 \zeta_{8} + \zeta_{8}^{2} ) q^{27} + ( -1 - \zeta_{8}^{2} ) q^{28} + ( -5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{29} + ( -3 + 2 \zeta_{8} + \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{30} -2 q^{31} -\zeta_{8} q^{32} + ( -1 - \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{33} + 2 \zeta_{8}^{2} q^{34} + ( 3 \zeta_{8} - \zeta_{8}^{3} ) q^{35} + ( -1 + 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{36} + ( 6 - 6 \zeta_{8}^{2} ) q^{37} -4 \zeta_{8}^{3} q^{38} + ( 2 + \zeta_{8}^{2} ) q^{40} + ( 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{41} + ( 1 + 2 \zeta_{8} - \zeta_{8}^{2} ) q^{42} + ( 6 + 6 \zeta_{8}^{2} ) q^{43} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{44} + ( -2 + \zeta_{8} - 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{45} + 4 q^{46} + ( 1 + \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{48} + 5 \zeta_{8}^{2} q^{49} + ( -4 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{50} + ( -2 - 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{51} + 4 \zeta_{8}^{3} q^{53} + ( -\zeta_{8} - 5 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{54} + ( 3 - \zeta_{8}^{2} ) q^{55} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{56} + ( -4 + 4 \zeta_{8} + 4 \zeta_{8}^{2} ) q^{57} + ( -5 - 5 \zeta_{8}^{2} ) q^{58} + ( 7 \zeta_{8} - 7 \zeta_{8}^{3} ) q^{59} + ( -1 - 3 \zeta_{8} + 2 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{60} -6 q^{61} -2 \zeta_{8} q^{62} + ( -1 - \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{63} -\zeta_{8}^{2} q^{64} + ( 2 - \zeta_{8} - \zeta_{8}^{3} ) q^{66} + ( -4 + 4 \zeta_{8}^{2} ) q^{67} + 2 \zeta_{8}^{3} q^{68} + ( -4 \zeta_{8} + 4 \zeta_{8}^{2} + 4 \zeta_{8}^{3} ) q^{69} + ( 1 + 3 \zeta_{8}^{2} ) q^{70} + ( -10 \zeta_{8} - 10 \zeta_{8}^{3} ) q^{71} + ( -2 - \zeta_{8} + 2 \zeta_{8}^{2} ) q^{72} + ( -5 - 5 \zeta_{8}^{2} ) q^{73} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{74} + ( 7 - 3 \zeta_{8} + \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{75} + 4 q^{76} -2 \zeta_{8} q^{77} -6 \zeta_{8}^{2} q^{79} + ( 2 \zeta_{8} + \zeta_{8}^{3} ) q^{80} + ( 7 + 4 \zeta_{8} + 4 \zeta_{8}^{3} ) q^{81} + ( -4 + 4 \zeta_{8}^{2} ) q^{82} + 12 \zeta_{8}^{3} q^{83} + ( \zeta_{8} + 2 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{84} + ( 2 - 4 \zeta_{8}^{2} ) q^{85} + ( 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{86} + ( 5 + 10 \zeta_{8} - 5 \zeta_{8}^{2} ) q^{87} + ( -1 - \zeta_{8}^{2} ) q^{88} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{89} + ( 2 - 2 \zeta_{8} + \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{90} + 4 \zeta_{8} q^{92} + ( 2 + 2 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{93} + ( -4 \zeta_{8} + 8 \zeta_{8}^{3} ) q^{95} + ( 1 + \zeta_{8} + \zeta_{8}^{3} ) q^{96} + ( 3 - 3 \zeta_{8}^{2} ) q^{97} + 5 \zeta_{8}^{3} q^{98} + ( -\zeta_{8} + 4 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{3} - 4q^{6} - 4q^{7} + O(q^{10})$$ $$4q - 4q^{3} - 4q^{6} - 4q^{7} + 4q^{10} + 4q^{12} + 8q^{15} - 4q^{16} + 8q^{18} + 8q^{21} - 4q^{22} - 16q^{25} - 4q^{27} - 4q^{28} - 12q^{30} - 8q^{31} - 4q^{33} - 4q^{36} + 24q^{37} + 8q^{40} + 4q^{42} + 24q^{43} - 8q^{45} + 16q^{46} + 4q^{48} - 8q^{51} + 12q^{55} - 16q^{57} - 20q^{58} - 4q^{60} - 24q^{61} - 4q^{63} + 8q^{66} - 16q^{67} + 4q^{70} - 8q^{72} - 20q^{73} + 28q^{75} + 16q^{76} + 28q^{81} - 16q^{82} + 8q^{85} + 20q^{87} - 4q^{88} + 8q^{90} + 8q^{93} + 4q^{96} + 12q^{97} + O(q^{100})$$

## Character values

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

 $$n$$ $$7$$ $$11$$ $$\chi(n)$$ $$-\zeta_{8}^{2}$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1
 −0.707107 + 0.707107i 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i
−0.707107 + 0.707107i −0.292893 + 1.70711i 1.00000i 0.707107 2.12132i −1.00000 1.41421i −1.00000 1.00000i 0.707107 + 0.707107i −2.82843 1.00000i 1.00000 + 2.00000i
17.2 0.707107 0.707107i −1.70711 + 0.292893i 1.00000i −0.707107 + 2.12132i −1.00000 + 1.41421i −1.00000 1.00000i −0.707107 0.707107i 2.82843 1.00000i 1.00000 + 2.00000i
23.1 −0.707107 0.707107i −0.292893 1.70711i 1.00000i 0.707107 + 2.12132i −1.00000 + 1.41421i −1.00000 + 1.00000i 0.707107 0.707107i −2.82843 + 1.00000i 1.00000 2.00000i
23.2 0.707107 + 0.707107i −1.70711 0.292893i 1.00000i −0.707107 2.12132i −1.00000 1.41421i −1.00000 + 1.00000i −0.707107 + 0.707107i 2.82843 + 1.00000i 1.00000 2.00000i
 $$n$$: 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
5.c odd 4 1 inner
15.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 30.2.e.a 4
3.b odd 2 1 inner 30.2.e.a 4
4.b odd 2 1 240.2.v.e 4
5.b even 2 1 150.2.e.a 4
5.c odd 4 1 inner 30.2.e.a 4
5.c odd 4 1 150.2.e.a 4
8.b even 2 1 960.2.v.k 4
8.d odd 2 1 960.2.v.c 4
9.c even 3 2 810.2.m.f 8
9.d odd 6 2 810.2.m.f 8
12.b even 2 1 240.2.v.e 4
15.d odd 2 1 150.2.e.a 4
15.e even 4 1 inner 30.2.e.a 4
15.e even 4 1 150.2.e.a 4
20.d odd 2 1 1200.2.v.b 4
20.e even 4 1 240.2.v.e 4
20.e even 4 1 1200.2.v.b 4
24.f even 2 1 960.2.v.c 4
24.h odd 2 1 960.2.v.k 4
40.i odd 4 1 960.2.v.k 4
40.k even 4 1 960.2.v.c 4
45.k odd 12 2 810.2.m.f 8
45.l even 12 2 810.2.m.f 8
60.h even 2 1 1200.2.v.b 4
60.l odd 4 1 240.2.v.e 4
60.l odd 4 1 1200.2.v.b 4
120.q odd 4 1 960.2.v.c 4
120.w even 4 1 960.2.v.k 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.2.e.a 4 1.a even 1 1 trivial
30.2.e.a 4 3.b odd 2 1 inner
30.2.e.a 4 5.c odd 4 1 inner
30.2.e.a 4 15.e even 4 1 inner
150.2.e.a 4 5.b even 2 1
150.2.e.a 4 5.c odd 4 1
150.2.e.a 4 15.d odd 2 1
150.2.e.a 4 15.e even 4 1
240.2.v.e 4 4.b odd 2 1
240.2.v.e 4 12.b even 2 1
240.2.v.e 4 20.e even 4 1
240.2.v.e 4 60.l odd 4 1
810.2.m.f 8 9.c even 3 2
810.2.m.f 8 9.d odd 6 2
810.2.m.f 8 45.k odd 12 2
810.2.m.f 8 45.l even 12 2
960.2.v.c 4 8.d odd 2 1
960.2.v.c 4 24.f even 2 1
960.2.v.c 4 40.k even 4 1
960.2.v.c 4 120.q odd 4 1
960.2.v.k 4 8.b even 2 1
960.2.v.k 4 24.h odd 2 1
960.2.v.k 4 40.i odd 4 1
960.2.v.k 4 120.w even 4 1
1200.2.v.b 4 20.d odd 2 1
1200.2.v.b 4 20.e even 4 1
1200.2.v.b 4 60.h even 2 1
1200.2.v.b 4 60.l odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(30, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T^{4}$$
$3$ $$1 + 4 T + 8 T^{2} + 12 T^{3} + 9 T^{4}$$
$5$ $$1 + 8 T^{2} + 25 T^{4}$$
$7$ $$( 1 + 2 T + 2 T^{2} + 14 T^{3} + 49 T^{4} )^{2}$$
$11$ $$( 1 - 20 T^{2} + 121 T^{4} )^{2}$$
$13$ $$( 1 + 169 T^{4} )^{2}$$
$17$ $$( 1 - 16 T^{2} + 289 T^{4} )( 1 + 16 T^{2} + 289 T^{4} )$$
$19$ $$( 1 - 22 T^{2} + 361 T^{4} )^{2}$$
$23$ $$1 - 158 T^{4} + 279841 T^{8}$$
$29$ $$( 1 + 8 T^{2} + 841 T^{4} )^{2}$$
$31$ $$( 1 + 2 T + 31 T^{2} )^{4}$$
$37$ $$( 1 - 12 T + 72 T^{2} - 444 T^{3} + 1369 T^{4} )^{2}$$
$41$ $$( 1 - 50 T^{2} + 1681 T^{4} )^{2}$$
$43$ $$( 1 - 12 T + 72 T^{2} - 516 T^{3} + 1849 T^{4} )^{2}$$
$47$ $$( 1 + 2209 T^{4} )^{2}$$
$53$ $$( 1 - 56 T^{2} + 2809 T^{4} )( 1 + 56 T^{2} + 2809 T^{4} )$$
$59$ $$( 1 + 20 T^{2} + 3481 T^{4} )^{2}$$
$61$ $$( 1 + 6 T + 61 T^{2} )^{4}$$
$67$ $$( 1 + 8 T + 32 T^{2} + 536 T^{3} + 4489 T^{4} )^{2}$$
$71$ $$( 1 + 58 T^{2} + 5041 T^{4} )^{2}$$
$73$ $$( 1 - 6 T + 73 T^{2} )^{2}( 1 + 16 T + 73 T^{2} )^{2}$$
$79$ $$( 1 - 122 T^{2} + 6241 T^{4} )^{2}$$
$83$ $$1 - 13294 T^{4} + 47458321 T^{8}$$
$89$ $$( 1 + 170 T^{2} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 6 T + 18 T^{2} - 582 T^{3} + 9409 T^{4} )^{2}$$