# Properties

 Label 30.3.f.a Level $30$ Weight $3$ Character orbit 30.f Analytic conductor $0.817$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$0.817440793081$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(i)$$ Coefficient field: $$\Q(i, \sqrt{6})$$ Defining polynomial: $$x^{4} + 9$$ x^4 + 9 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{2} + 1) q^{2} + \beta_1 q^{3} - 2 \beta_{2} q^{4} + (2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{5} + ( - \beta_{3} + \beta_1) q^{6} + ( - 4 \beta_{3} + 4 \beta_{2} - 4) q^{7} + ( - 2 \beta_{2} - 2) q^{8} + 3 \beta_{2} q^{9}+O(q^{10})$$ q + (-b2 + 1) * q^2 + b1 * q^3 - 2*b2 * q^4 + (2*b3 + b2 - 2*b1) * q^5 + (-b3 + b1) * q^6 + (-4*b3 + 4*b2 - 4) * q^7 + (-2*b2 - 2) * q^8 + 3*b2 * q^9 $$q + ( - \beta_{2} + 1) q^{2} + \beta_1 q^{3} - 2 \beta_{2} q^{4} + (2 \beta_{3} + \beta_{2} - 2 \beta_1) q^{5} + ( - \beta_{3} + \beta_1) q^{6} + ( - 4 \beta_{3} + 4 \beta_{2} - 4) q^{7} + ( - 2 \beta_{2} - 2) q^{8} + 3 \beta_{2} q^{9} + (4 \beta_{3} + \beta_{2} + 1) q^{10} + (4 \beta_{3} - 4 \beta_1 - 4) q^{11} - 2 \beta_{3} q^{12} + (3 \beta_{2} + 8 \beta_1 + 3) q^{13} + ( - 4 \beta_{3} + 8 \beta_{2} - 4 \beta_1) q^{14} + (\beta_{3} - 6 \beta_{2} - 6) q^{15} - 4 q^{16} + ( - 4 \beta_{3} - 11 \beta_{2} + 11) q^{17} + (3 \beta_{2} + 3) q^{18} + ( - 4 \beta_{3} - 16 \beta_{2} - 4 \beta_1) q^{19} + (4 \beta_{3} + 4 \beta_1 + 2) q^{20} + (4 \beta_{3} - 4 \beta_1 + 12) q^{21} + (8 \beta_{3} + 4 \beta_{2} - 4) q^{22} + ( - 4 \beta_{2} + 12 \beta_1 - 4) q^{23} + ( - 2 \beta_{3} - 2 \beta_1) q^{24} + ( - 4 \beta_{3} - 4 \beta_1 + 23) q^{25} + ( - 8 \beta_{3} + 8 \beta_1 + 6) q^{26} + 3 \beta_{3} q^{27} + (8 \beta_{2} - 8 \beta_1 + 8) q^{28} + (4 \beta_{3} + 16 \beta_{2} + 4 \beta_1) q^{29} + (\beta_{3} + \beta_1 - 12) q^{30} + (8 \beta_{3} - 8 \beta_1 - 20) q^{31} + (4 \beta_{2} - 4) q^{32} + ( - 12 \beta_{2} - 4 \beta_1 - 12) q^{33} + ( - 4 \beta_{3} - 22 \beta_{2} - 4 \beta_1) q^{34} + ( - 16 \beta_{3} + 20 \beta_{2} + 4 \beta_1 - 28) q^{35} + 6 q^{36} + (27 \beta_{2} - 27) q^{37} + ( - 16 \beta_{2} - 8 \beta_1 - 16) q^{38} + (3 \beta_{3} + 24 \beta_{2} + 3 \beta_1) q^{39} + ( - 2 \beta_{2} + 8 \beta_1 + 2) q^{40} + ( - 4 \beta_{3} + 4 \beta_1 + 8) q^{41} + (8 \beta_{3} - 12 \beta_{2} + 12) q^{42} + (12 \beta_{2} - 20 \beta_1 + 12) q^{43} + (8 \beta_{3} + 8 \beta_{2} + 8 \beta_1) q^{44} + ( - 6 \beta_{3} - 6 \beta_1 - 3) q^{45} + ( - 12 \beta_{3} + 12 \beta_1 - 8) q^{46} + (12 \beta_{3} - 24 \beta_{2} + 24) q^{47} - 4 \beta_1 q^{48} + (32 \beta_{3} - 31 \beta_{2} + 32 \beta_1) q^{49} + ( - 23 \beta_{2} - 8 \beta_1 + 23) q^{50} + ( - 11 \beta_{3} + 11 \beta_1 + 12) q^{51} + ( - 16 \beta_{3} - 6 \beta_{2} + 6) q^{52} + (25 \beta_{2} - 36 \beta_1 + 25) q^{53} + (3 \beta_{3} + 3 \beta_1) q^{54} + ( - 12 \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 48) q^{55} + (8 \beta_{3} - 8 \beta_1 + 16) q^{56} + ( - 16 \beta_{3} - 12 \beta_{2} + 12) q^{57} + (16 \beta_{2} + 8 \beta_1 + 16) q^{58} - 20 \beta_{2} q^{59} + (12 \beta_{2} + 2 \beta_1 - 12) q^{60} + ( - 16 \beta_{3} + 16 \beta_1 - 24) q^{61} + (16 \beta_{3} + 20 \beta_{2} - 20) q^{62} + ( - 12 \beta_{2} + 12 \beta_1 - 12) q^{63} + 8 \beta_{2} q^{64} + (8 \beta_{3} - 45 \beta_{2} - 12 \beta_1 - 51) q^{65} + (4 \beta_{3} - 4 \beta_1 - 24) q^{66} + (36 \beta_{3} + 4 \beta_{2} - 4) q^{67} + ( - 22 \beta_{2} - 8 \beta_1 - 22) q^{68} + ( - 4 \beta_{3} + 36 \beta_{2} - 4 \beta_1) q^{69} + ( - 20 \beta_{3} + 48 \beta_{2} - 12 \beta_1 - 8) q^{70} + (4 \beta_{3} - 4 \beta_1 + 16) q^{71} + ( - 6 \beta_{2} + 6) q^{72} + ( - 47 \beta_{2} + 8 \beta_1 - 47) q^{73} + 54 \beta_{2} q^{74} + ( - 12 \beta_{2} + 23 \beta_1 + 12) q^{75} + (8 \beta_{3} - 8 \beta_1 - 32) q^{76} + ( - 16 \beta_{3} + 32 \beta_{2} - 32) q^{77} + (24 \beta_{2} + 6 \beta_1 + 24) q^{78} + ( - 52 \beta_{3} + 12 \beta_{2} - 52 \beta_1) q^{79} + ( - 8 \beta_{3} - 4 \beta_{2} + 8 \beta_1) q^{80} - 9 q^{81} + ( - 8 \beta_{3} - 8 \beta_{2} + 8) q^{82} + (48 \beta_{2} + 28 \beta_1 + 48) q^{83} + (8 \beta_{3} - 24 \beta_{2} + 8 \beta_1) q^{84} + (44 \beta_{3} + 35 \beta_{2} + 4 \beta_1 - 13) q^{85} + (20 \beta_{3} - 20 \beta_1 + 24) q^{86} + (16 \beta_{3} + 12 \beta_{2} - 12) q^{87} + (8 \beta_{2} + 16 \beta_1 + 8) q^{88} + ( - 12 \beta_{3} - 88 \beta_{2} - 12 \beta_1) q^{89} + (3 \beta_{2} - 12 \beta_1 - 3) q^{90} + (20 \beta_{3} - 20 \beta_1 + 72) q^{91} + ( - 24 \beta_{3} + 8 \beta_{2} - 8) q^{92} + ( - 24 \beta_{2} - 20 \beta_1 - 24) q^{93} + (12 \beta_{3} - 48 \beta_{2} + 12 \beta_1) q^{94} + (28 \beta_{3} + 48 \beta_{2} + 36 \beta_1 + 16) q^{95} + (4 \beta_{3} - 4 \beta_1) q^{96} + ( - 40 \beta_{3} - 33 \beta_{2} + 33) q^{97} + ( - 31 \beta_{2} + 64 \beta_1 - 31) q^{98} + ( - 12 \beta_{3} - 12 \beta_{2} - 12 \beta_1) q^{99}+O(q^{100})$$ q + (-b2 + 1) * q^2 + b1 * q^3 - 2*b2 * q^4 + (2*b3 + b2 - 2*b1) * q^5 + (-b3 + b1) * q^6 + (-4*b3 + 4*b2 - 4) * q^7 + (-2*b2 - 2) * q^8 + 3*b2 * q^9 + (4*b3 + b2 + 1) * q^10 + (4*b3 - 4*b1 - 4) * q^11 - 2*b3 * q^12 + (3*b2 + 8*b1 + 3) * q^13 + (-4*b3 + 8*b2 - 4*b1) * q^14 + (b3 - 6*b2 - 6) * q^15 - 4 * q^16 + (-4*b3 - 11*b2 + 11) * q^17 + (3*b2 + 3) * q^18 + (-4*b3 - 16*b2 - 4*b1) * q^19 + (4*b3 + 4*b1 + 2) * q^20 + (4*b3 - 4*b1 + 12) * q^21 + (8*b3 + 4*b2 - 4) * q^22 + (-4*b2 + 12*b1 - 4) * q^23 + (-2*b3 - 2*b1) * q^24 + (-4*b3 - 4*b1 + 23) * q^25 + (-8*b3 + 8*b1 + 6) * q^26 + 3*b3 * q^27 + (8*b2 - 8*b1 + 8) * q^28 + (4*b3 + 16*b2 + 4*b1) * q^29 + (b3 + b1 - 12) * q^30 + (8*b3 - 8*b1 - 20) * q^31 + (4*b2 - 4) * q^32 + (-12*b2 - 4*b1 - 12) * q^33 + (-4*b3 - 22*b2 - 4*b1) * q^34 + (-16*b3 + 20*b2 + 4*b1 - 28) * q^35 + 6 * q^36 + (27*b2 - 27) * q^37 + (-16*b2 - 8*b1 - 16) * q^38 + (3*b3 + 24*b2 + 3*b1) * q^39 + (-2*b2 + 8*b1 + 2) * q^40 + (-4*b3 + 4*b1 + 8) * q^41 + (8*b3 - 12*b2 + 12) * q^42 + (12*b2 - 20*b1 + 12) * q^43 + (8*b3 + 8*b2 + 8*b1) * q^44 + (-6*b3 - 6*b1 - 3) * q^45 + (-12*b3 + 12*b1 - 8) * q^46 + (12*b3 - 24*b2 + 24) * q^47 - 4*b1 * q^48 + (32*b3 - 31*b2 + 32*b1) * q^49 + (-23*b2 - 8*b1 + 23) * q^50 + (-11*b3 + 11*b1 + 12) * q^51 + (-16*b3 - 6*b2 + 6) * q^52 + (25*b2 - 36*b1 + 25) * q^53 + (3*b3 + 3*b1) * q^54 + (-12*b3 - 4*b2 + 4*b1 + 48) * q^55 + (8*b3 - 8*b1 + 16) * q^56 + (-16*b3 - 12*b2 + 12) * q^57 + (16*b2 + 8*b1 + 16) * q^58 - 20*b2 * q^59 + (12*b2 + 2*b1 - 12) * q^60 + (-16*b3 + 16*b1 - 24) * q^61 + (16*b3 + 20*b2 - 20) * q^62 + (-12*b2 + 12*b1 - 12) * q^63 + 8*b2 * q^64 + (8*b3 - 45*b2 - 12*b1 - 51) * q^65 + (4*b3 - 4*b1 - 24) * q^66 + (36*b3 + 4*b2 - 4) * q^67 + (-22*b2 - 8*b1 - 22) * q^68 + (-4*b3 + 36*b2 - 4*b1) * q^69 + (-20*b3 + 48*b2 - 12*b1 - 8) * q^70 + (4*b3 - 4*b1 + 16) * q^71 + (-6*b2 + 6) * q^72 + (-47*b2 + 8*b1 - 47) * q^73 + 54*b2 * q^74 + (-12*b2 + 23*b1 + 12) * q^75 + (8*b3 - 8*b1 - 32) * q^76 + (-16*b3 + 32*b2 - 32) * q^77 + (24*b2 + 6*b1 + 24) * q^78 + (-52*b3 + 12*b2 - 52*b1) * q^79 + (-8*b3 - 4*b2 + 8*b1) * q^80 - 9 * q^81 + (-8*b3 - 8*b2 + 8) * q^82 + (48*b2 + 28*b1 + 48) * q^83 + (8*b3 - 24*b2 + 8*b1) * q^84 + (44*b3 + 35*b2 + 4*b1 - 13) * q^85 + (20*b3 - 20*b1 + 24) * q^86 + (16*b3 + 12*b2 - 12) * q^87 + (8*b2 + 16*b1 + 8) * q^88 + (-12*b3 - 88*b2 - 12*b1) * q^89 + (3*b2 - 12*b1 - 3) * q^90 + (20*b3 - 20*b1 + 72) * q^91 + (-24*b3 + 8*b2 - 8) * q^92 + (-24*b2 - 20*b1 - 24) * q^93 + (12*b3 - 48*b2 + 12*b1) * q^94 + (28*b3 + 48*b2 + 36*b1 + 16) * q^95 + (4*b3 - 4*b1) * q^96 + (-40*b3 - 33*b2 + 33) * q^97 + (-31*b2 + 64*b1 - 31) * q^98 + (-12*b3 - 12*b2 - 12*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} - 16 q^{7} - 8 q^{8}+O(q^{10})$$ 4 * q + 4 * q^2 - 16 * q^7 - 8 * q^8 $$4 q + 4 q^{2} - 16 q^{7} - 8 q^{8} + 4 q^{10} - 16 q^{11} + 12 q^{13} - 24 q^{15} - 16 q^{16} + 44 q^{17} + 12 q^{18} + 8 q^{20} + 48 q^{21} - 16 q^{22} - 16 q^{23} + 92 q^{25} + 24 q^{26} + 32 q^{28} - 48 q^{30} - 80 q^{31} - 16 q^{32} - 48 q^{33} - 112 q^{35} + 24 q^{36} - 108 q^{37} - 64 q^{38} + 8 q^{40} + 32 q^{41} + 48 q^{42} + 48 q^{43} - 12 q^{45} - 32 q^{46} + 96 q^{47} + 92 q^{50} + 48 q^{51} + 24 q^{52} + 100 q^{53} + 192 q^{55} + 64 q^{56} + 48 q^{57} + 64 q^{58} - 48 q^{60} - 96 q^{61} - 80 q^{62} - 48 q^{63} - 204 q^{65} - 96 q^{66} - 16 q^{67} - 88 q^{68} - 32 q^{70} + 64 q^{71} + 24 q^{72} - 188 q^{73} + 48 q^{75} - 128 q^{76} - 128 q^{77} + 96 q^{78} - 36 q^{81} + 32 q^{82} + 192 q^{83} - 52 q^{85} + 96 q^{86} - 48 q^{87} + 32 q^{88} - 12 q^{90} + 288 q^{91} - 32 q^{92} - 96 q^{93} + 64 q^{95} + 132 q^{97} - 124 q^{98}+O(q^{100})$$ 4 * q + 4 * q^2 - 16 * q^7 - 8 * q^8 + 4 * q^10 - 16 * q^11 + 12 * q^13 - 24 * q^15 - 16 * q^16 + 44 * q^17 + 12 * q^18 + 8 * q^20 + 48 * q^21 - 16 * q^22 - 16 * q^23 + 92 * q^25 + 24 * q^26 + 32 * q^28 - 48 * q^30 - 80 * q^31 - 16 * q^32 - 48 * q^33 - 112 * q^35 + 24 * q^36 - 108 * q^37 - 64 * q^38 + 8 * q^40 + 32 * q^41 + 48 * q^42 + 48 * q^43 - 12 * q^45 - 32 * q^46 + 96 * q^47 + 92 * q^50 + 48 * q^51 + 24 * q^52 + 100 * q^53 + 192 * q^55 + 64 * q^56 + 48 * q^57 + 64 * q^58 - 48 * q^60 - 96 * q^61 - 80 * q^62 - 48 * q^63 - 204 * q^65 - 96 * q^66 - 16 * q^67 - 88 * q^68 - 32 * q^70 + 64 * q^71 + 24 * q^72 - 188 * q^73 + 48 * q^75 - 128 * q^76 - 128 * q^77 + 96 * q^78 - 36 * q^81 + 32 * q^82 + 192 * q^83 - 52 * q^85 + 96 * q^86 - 48 * q^87 + 32 * q^88 - 12 * q^90 + 288 * q^91 - 32 * q^92 - 96 * q^93 + 64 * q^95 + 132 * q^97 - 124 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{2} ) / 3$$ (v^2) / 3 $$\beta_{3}$$ $$=$$ $$( \nu^{3} ) / 3$$ (v^3) / 3
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$3\beta_{2}$$ 3*b2 $$\nu^{3}$$ $$=$$ $$3\beta_{3}$$ 3*b3

## Character values

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

 $$n$$ $$7$$ $$11$$ $$\chi(n)$$ $$-\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
7.1
 −1.22474 + 1.22474i 1.22474 − 1.22474i −1.22474 − 1.22474i 1.22474 + 1.22474i
1.00000 + 1.00000i −1.22474 + 1.22474i 2.00000i 4.89898 1.00000i −2.44949 −8.89898 8.89898i −2.00000 + 2.00000i 3.00000i 5.89898 + 3.89898i
7.2 1.00000 + 1.00000i 1.22474 1.22474i 2.00000i −4.89898 1.00000i 2.44949 0.898979 + 0.898979i −2.00000 + 2.00000i 3.00000i −3.89898 5.89898i
13.1 1.00000 1.00000i −1.22474 1.22474i 2.00000i 4.89898 + 1.00000i −2.44949 −8.89898 + 8.89898i −2.00000 2.00000i 3.00000i 5.89898 3.89898i
13.2 1.00000 1.00000i 1.22474 + 1.22474i 2.00000i −4.89898 + 1.00000i 2.44949 0.898979 0.898979i −2.00000 2.00000i 3.00000i −3.89898 + 5.89898i
 $$n$$: 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.c odd 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 30.3.f.a 4
3.b odd 2 1 90.3.g.d 4
4.b odd 2 1 240.3.bg.b 4
5.b even 2 1 150.3.f.b 4
5.c odd 4 1 inner 30.3.f.a 4
5.c odd 4 1 150.3.f.b 4
8.b even 2 1 960.3.bg.e 4
8.d odd 2 1 960.3.bg.g 4
12.b even 2 1 720.3.bh.i 4
15.d odd 2 1 450.3.g.j 4
15.e even 4 1 90.3.g.d 4
15.e even 4 1 450.3.g.j 4
20.d odd 2 1 1200.3.bg.d 4
20.e even 4 1 240.3.bg.b 4
20.e even 4 1 1200.3.bg.d 4
40.i odd 4 1 960.3.bg.e 4
40.k even 4 1 960.3.bg.g 4
60.l odd 4 1 720.3.bh.i 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.f.a 4 1.a even 1 1 trivial
30.3.f.a 4 5.c odd 4 1 inner
90.3.g.d 4 3.b odd 2 1
90.3.g.d 4 15.e even 4 1
150.3.f.b 4 5.b even 2 1
150.3.f.b 4 5.c odd 4 1
240.3.bg.b 4 4.b odd 2 1
240.3.bg.b 4 20.e even 4 1
450.3.g.j 4 15.d odd 2 1
450.3.g.j 4 15.e even 4 1
720.3.bh.i 4 12.b even 2 1
720.3.bh.i 4 60.l odd 4 1
960.3.bg.e 4 8.b even 2 1
960.3.bg.e 4 40.i odd 4 1
960.3.bg.g 4 8.d odd 2 1
960.3.bg.g 4 40.k even 4 1
1200.3.bg.d 4 20.d odd 2 1
1200.3.bg.d 4 20.e even 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(30, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} - 2 T + 2)^{2}$$
$3$ $$T^{4} + 9$$
$5$ $$T^{4} - 46T^{2} + 625$$
$7$ $$T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 256$$
$11$ $$(T^{2} + 8 T - 80)^{2}$$
$13$ $$T^{4} - 12 T^{3} + 72 T^{2} + \cdots + 30276$$
$17$ $$T^{4} - 44 T^{3} + 968 T^{2} + \cdots + 37636$$
$19$ $$T^{4} + 704 T^{2} + 25600$$
$23$ $$T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 160000$$
$29$ $$T^{4} + 704 T^{2} + 25600$$
$31$ $$(T^{2} + 40 T + 16)^{2}$$
$37$ $$(T^{2} + 54 T + 1458)^{2}$$
$41$ $$(T^{2} - 16 T - 32)^{2}$$
$43$ $$T^{4} - 48 T^{3} + 1152 T^{2} + \cdots + 831744$$
$47$ $$T^{4} - 96 T^{3} + 4608 T^{2} + \cdots + 518400$$
$53$ $$T^{4} - 100 T^{3} + 5000 T^{2} + \cdots + 6959044$$
$59$ $$(T^{2} + 400)^{2}$$
$61$ $$(T^{2} + 48 T - 960)^{2}$$
$67$ $$T^{4} + 16 T^{3} + 128 T^{2} + \cdots + 14868736$$
$71$ $$(T^{2} - 32 T + 160)^{2}$$
$73$ $$T^{4} + 188 T^{3} + \cdots + 17859076$$
$79$ $$T^{4} + 32736 T^{2} + \cdots + 258566400$$
$83$ $$T^{4} - 192 T^{3} + 18432 T^{2} + \cdots + 5089536$$
$89$ $$T^{4} + 17216 T^{2} + \cdots + 47334400$$
$97$ $$T^{4} - 132 T^{3} + 8712 T^{2} + \cdots + 6874884$$