# Properties

 Label 30.3.d.a Level $30$ Weight $3$ Character orbit 30.d 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.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.817440793081$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-2}, \sqrt{-5})$$ Defining polynomial: $$x^{4} - 4x^{2} + 9$$ x^4 - 4*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 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.

 $$f(q)$$ $$=$$ $$q - \beta_{2} q^{2} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{3} - 2 q^{4} - \beta_{3} q^{5} + ( - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{6} + (3 \beta_{2} - 6 \beta_1 + 2) q^{7} + 2 \beta_{2} q^{8} + (\beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{9}+O(q^{10})$$ q - b2 * q^2 + (b3 - b2 + b1 + 1) * q^3 - 2 * q^4 - b3 * q^5 + (-b3 - 2*b2 + 2*b1 - 1) * q^6 + (3*b2 - 6*b1 + 2) * q^7 + 2*b2 * q^8 + (b3 + 2*b2 + 4*b1 - 2) * q^9 $$q - \beta_{2} q^{2} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{3} - 2 q^{4} - \beta_{3} q^{5} + ( - \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 1) q^{6} + (3 \beta_{2} - 6 \beta_1 + 2) q^{7} + 2 \beta_{2} q^{8} + (\beta_{3} + 2 \beta_{2} + 4 \beta_1 - 2) q^{9} + (\beta_{2} - 2 \beta_1) q^{10} + 6 \beta_{2} q^{11} + ( - 2 \beta_{3} + 2 \beta_{2} - 2 \beta_1 - 2) q^{12} - 10 q^{13} + (6 \beta_{3} - 2 \beta_{2}) q^{14} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 5) q^{15} + 4 q^{16} + ( - 6 \beta_{3} - 12 \beta_{2}) q^{17} + ( - 4 \beta_{3} + \beta_{2} + 2 \beta_1 + 8) q^{18} + ( - 6 \beta_{2} + 12 \beta_1 + 8) q^{19} + 2 \beta_{3} q^{20} + (5 \beta_{3} - 14 \beta_{2} - 4 \beta_1 - 13) q^{21} + 12 q^{22} + (6 \beta_{3} - 3 \beta_{2}) q^{23} + (2 \beta_{3} + 4 \beta_{2} - 4 \beta_1 + 2) q^{24} - 5 q^{25} + 10 \beta_{2} q^{26} + (\beta_{3} + 20 \beta_{2} - 5 \beta_1 + 7) q^{27} + ( - 6 \beta_{2} + 12 \beta_1 - 4) q^{28} - 12 \beta_{3} q^{29} + (\beta_{3} - 4 \beta_{2} - 2 \beta_1 - 5) q^{30} + 8 q^{31} - 4 \beta_{2} q^{32} + (6 \beta_{3} + 12 \beta_{2} - 12 \beta_1 + 6) q^{33} + (6 \beta_{2} - 12 \beta_1 - 24) q^{34} + ( - 2 \beta_{3} + 15 \beta_{2}) q^{35} + ( - 2 \beta_{3} - 4 \beta_{2} - 8 \beta_1 + 4) q^{36} + ( - 12 \beta_{2} + 24 \beta_1 - 22) q^{37} + ( - 12 \beta_{3} - 8 \beta_{2}) q^{38} + ( - 10 \beta_{3} + 10 \beta_{2} - 10 \beta_1 - 10) q^{39} + ( - 2 \beta_{2} + 4 \beta_1) q^{40} + (6 \beta_{3} + 24 \beta_{2}) q^{41} + (4 \beta_{3} + 8 \beta_{2} + 10 \beta_1 - 32) q^{42} + (9 \beta_{2} - 18 \beta_1 + 14) q^{43} - 12 \beta_{2} q^{44} + (2 \beta_{3} - 14 \beta_{2} + 8 \beta_1 + 5) q^{45} + ( - 6 \beta_{2} + 12 \beta_1 - 6) q^{46} + (30 \beta_{3} - 15 \beta_{2}) q^{47} + (4 \beta_{3} - 4 \beta_{2} + 4 \beta_1 + 4) q^{48} + (12 \beta_{2} - 24 \beta_1 + 45) q^{49} + 5 \beta_{2} q^{50} + ( - 18 \beta_{3} - 36 \beta_{2} + 18 \beta_1 + 18) q^{51} + 20 q^{52} + ( - 6 \beta_{3} - 12 \beta_{2}) q^{53} + (5 \beta_{3} - 8 \beta_{2} + 2 \beta_1 + 35) q^{54} + ( - 6 \beta_{2} + 12 \beta_1) q^{55} + ( - 12 \beta_{3} + 4 \beta_{2}) q^{56} + (2 \beta_{3} + 16 \beta_{2} + 20 \beta_1 + 38) q^{57} + (12 \beta_{2} - 24 \beta_1) q^{58} + (12 \beta_{3} - 36 \beta_{2}) q^{59} + (2 \beta_{3} + 4 \beta_{2} + 2 \beta_1 - 10) q^{60} + (12 \beta_{2} - 24 \beta_1 - 16) q^{61} - 8 \beta_{2} q^{62} + ( - 22 \beta_{3} - 17 \beta_{2} + 20 \beta_1 - 64) q^{63} - 8 q^{64} + 10 \beta_{3} q^{65} + (12 \beta_{3} - 12 \beta_{2} + 12 \beta_1 + 12) q^{66} + (9 \beta_{2} - 18 \beta_1 - 82) q^{67} + (12 \beta_{3} + 24 \beta_{2}) q^{68} + (3 \beta_{3} + 6 \beta_{2} + 12 \beta_1 - 33) q^{69} + (2 \beta_{2} - 4 \beta_1 + 30) q^{70} + ( - 12 \beta_{3} + 30 \beta_{2}) q^{71} + (8 \beta_{3} - 2 \beta_{2} - 4 \beta_1 - 16) q^{72} + ( - 12 \beta_{2} + 24 \beta_1 + 50) q^{73} + ( - 24 \beta_{3} + 22 \beta_{2}) q^{74} + ( - 5 \beta_{3} + 5 \beta_{2} - 5 \beta_1 - 5) q^{75} + (12 \beta_{2} - 24 \beta_1 - 16) q^{76} + ( - 36 \beta_{3} + 12 \beta_{2}) q^{77} + (10 \beta_{3} + 20 \beta_{2} - 20 \beta_1 + 10) q^{78} + (6 \beta_{2} - 12 \beta_1 - 28) q^{79} - 4 \beta_{3} q^{80} + (28 \beta_{3} + 20 \beta_{2} - 32 \beta_1 + 7) q^{81} + ( - 6 \beta_{2} + 12 \beta_1 + 48) q^{82} + (6 \beta_{3} + 9 \beta_{2}) q^{83} + ( - 10 \beta_{3} + 28 \beta_{2} + 8 \beta_1 + 26) q^{84} + (12 \beta_{2} - 24 \beta_1 - 30) q^{85} + (18 \beta_{3} - 14 \beta_{2}) q^{86} + ( - 12 \beta_{3} - 24 \beta_{2} - 12 \beta_1 + 60) q^{87} - 24 q^{88} + (12 \beta_{3} + 24 \beta_{2}) q^{89} + ( - 8 \beta_{3} - 7 \beta_{2} + 4 \beta_1 - 20) q^{90} + ( - 30 \beta_{2} + 60 \beta_1 - 20) q^{91} + ( - 12 \beta_{3} + 6 \beta_{2}) q^{92} + (8 \beta_{3} - 8 \beta_{2} + 8 \beta_1 + 8) q^{93} + ( - 30 \beta_{2} + 60 \beta_1 - 30) q^{94} + ( - 8 \beta_{3} - 30 \beta_{2}) q^{95} + ( - 4 \beta_{3} - 8 \beta_{2} + 8 \beta_1 - 4) q^{96} + (12 \beta_{2} - 24 \beta_1 + 74) q^{97} + (24 \beta_{3} - 45 \beta_{2}) q^{98} + (24 \beta_{3} - 6 \beta_{2} - 12 \beta_1 - 48) q^{99}+O(q^{100})$$ q - b2 * q^2 + (b3 - b2 + b1 + 1) * q^3 - 2 * q^4 - b3 * q^5 + (-b3 - 2*b2 + 2*b1 - 1) * q^6 + (3*b2 - 6*b1 + 2) * q^7 + 2*b2 * q^8 + (b3 + 2*b2 + 4*b1 - 2) * q^9 + (b2 - 2*b1) * q^10 + 6*b2 * q^11 + (-2*b3 + 2*b2 - 2*b1 - 2) * q^12 - 10 * q^13 + (6*b3 - 2*b2) * q^14 + (-b3 - 2*b2 - b1 + 5) * q^15 + 4 * q^16 + (-6*b3 - 12*b2) * q^17 + (-4*b3 + b2 + 2*b1 + 8) * q^18 + (-6*b2 + 12*b1 + 8) * q^19 + 2*b3 * q^20 + (5*b3 - 14*b2 - 4*b1 - 13) * q^21 + 12 * q^22 + (6*b3 - 3*b2) * q^23 + (2*b3 + 4*b2 - 4*b1 + 2) * q^24 - 5 * q^25 + 10*b2 * q^26 + (b3 + 20*b2 - 5*b1 + 7) * q^27 + (-6*b2 + 12*b1 - 4) * q^28 - 12*b3 * q^29 + (b3 - 4*b2 - 2*b1 - 5) * q^30 + 8 * q^31 - 4*b2 * q^32 + (6*b3 + 12*b2 - 12*b1 + 6) * q^33 + (6*b2 - 12*b1 - 24) * q^34 + (-2*b3 + 15*b2) * q^35 + (-2*b3 - 4*b2 - 8*b1 + 4) * q^36 + (-12*b2 + 24*b1 - 22) * q^37 + (-12*b3 - 8*b2) * q^38 + (-10*b3 + 10*b2 - 10*b1 - 10) * q^39 + (-2*b2 + 4*b1) * q^40 + (6*b3 + 24*b2) * q^41 + (4*b3 + 8*b2 + 10*b1 - 32) * q^42 + (9*b2 - 18*b1 + 14) * q^43 - 12*b2 * q^44 + (2*b3 - 14*b2 + 8*b1 + 5) * q^45 + (-6*b2 + 12*b1 - 6) * q^46 + (30*b3 - 15*b2) * q^47 + (4*b3 - 4*b2 + 4*b1 + 4) * q^48 + (12*b2 - 24*b1 + 45) * q^49 + 5*b2 * q^50 + (-18*b3 - 36*b2 + 18*b1 + 18) * q^51 + 20 * q^52 + (-6*b3 - 12*b2) * q^53 + (5*b3 - 8*b2 + 2*b1 + 35) * q^54 + (-6*b2 + 12*b1) * q^55 + (-12*b3 + 4*b2) * q^56 + (2*b3 + 16*b2 + 20*b1 + 38) * q^57 + (12*b2 - 24*b1) * q^58 + (12*b3 - 36*b2) * q^59 + (2*b3 + 4*b2 + 2*b1 - 10) * q^60 + (12*b2 - 24*b1 - 16) * q^61 - 8*b2 * q^62 + (-22*b3 - 17*b2 + 20*b1 - 64) * q^63 - 8 * q^64 + 10*b3 * q^65 + (12*b3 - 12*b2 + 12*b1 + 12) * q^66 + (9*b2 - 18*b1 - 82) * q^67 + (12*b3 + 24*b2) * q^68 + (3*b3 + 6*b2 + 12*b1 - 33) * q^69 + (2*b2 - 4*b1 + 30) * q^70 + (-12*b3 + 30*b2) * q^71 + (8*b3 - 2*b2 - 4*b1 - 16) * q^72 + (-12*b2 + 24*b1 + 50) * q^73 + (-24*b3 + 22*b2) * q^74 + (-5*b3 + 5*b2 - 5*b1 - 5) * q^75 + (12*b2 - 24*b1 - 16) * q^76 + (-36*b3 + 12*b2) * q^77 + (10*b3 + 20*b2 - 20*b1 + 10) * q^78 + (6*b2 - 12*b1 - 28) * q^79 - 4*b3 * q^80 + (28*b3 + 20*b2 - 32*b1 + 7) * q^81 + (-6*b2 + 12*b1 + 48) * q^82 + (6*b3 + 9*b2) * q^83 + (-10*b3 + 28*b2 + 8*b1 + 26) * q^84 + (12*b2 - 24*b1 - 30) * q^85 + (18*b3 - 14*b2) * q^86 + (-12*b3 - 24*b2 - 12*b1 + 60) * q^87 - 24 * q^88 + (12*b3 + 24*b2) * q^89 + (-8*b3 - 7*b2 + 4*b1 - 20) * q^90 + (-30*b2 + 60*b1 - 20) * q^91 + (-12*b3 + 6*b2) * q^92 + (8*b3 - 8*b2 + 8*b1 + 8) * q^93 + (-30*b2 + 60*b1 - 30) * q^94 + (-8*b3 - 30*b2) * q^95 + (-4*b3 - 8*b2 + 8*b1 - 4) * q^96 + (12*b2 - 24*b1 + 74) * q^97 + (24*b3 - 45*b2) * q^98 + (24*b3 - 6*b2 - 12*b1 - 48) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{3} - 8 q^{4} - 4 q^{6} + 8 q^{7} - 8 q^{9}+O(q^{10})$$ 4 * q + 4 * q^3 - 8 * q^4 - 4 * q^6 + 8 * q^7 - 8 * q^9 $$4 q + 4 q^{3} - 8 q^{4} - 4 q^{6} + 8 q^{7} - 8 q^{9} - 8 q^{12} - 40 q^{13} + 20 q^{15} + 16 q^{16} + 32 q^{18} + 32 q^{19} - 52 q^{21} + 48 q^{22} + 8 q^{24} - 20 q^{25} + 28 q^{27} - 16 q^{28} - 20 q^{30} + 32 q^{31} + 24 q^{33} - 96 q^{34} + 16 q^{36} - 88 q^{37} - 40 q^{39} - 128 q^{42} + 56 q^{43} + 20 q^{45} - 24 q^{46} + 16 q^{48} + 180 q^{49} + 72 q^{51} + 80 q^{52} + 140 q^{54} + 152 q^{57} - 40 q^{60} - 64 q^{61} - 256 q^{63} - 32 q^{64} + 48 q^{66} - 328 q^{67} - 132 q^{69} + 120 q^{70} - 64 q^{72} + 200 q^{73} - 20 q^{75} - 64 q^{76} + 40 q^{78} - 112 q^{79} + 28 q^{81} + 192 q^{82} + 104 q^{84} - 120 q^{85} + 240 q^{87} - 96 q^{88} - 80 q^{90} - 80 q^{91} + 32 q^{93} - 120 q^{94} - 16 q^{96} + 296 q^{97} - 192 q^{99}+O(q^{100})$$ 4 * q + 4 * q^3 - 8 * q^4 - 4 * q^6 + 8 * q^7 - 8 * q^9 - 8 * q^12 - 40 * q^13 + 20 * q^15 + 16 * q^16 + 32 * q^18 + 32 * q^19 - 52 * q^21 + 48 * q^22 + 8 * q^24 - 20 * q^25 + 28 * q^27 - 16 * q^28 - 20 * q^30 + 32 * q^31 + 24 * q^33 - 96 * q^34 + 16 * q^36 - 88 * q^37 - 40 * q^39 - 128 * q^42 + 56 * q^43 + 20 * q^45 - 24 * q^46 + 16 * q^48 + 180 * q^49 + 72 * q^51 + 80 * q^52 + 140 * q^54 + 152 * q^57 - 40 * q^60 - 64 * q^61 - 256 * q^63 - 32 * q^64 + 48 * q^66 - 328 * q^67 - 132 * q^69 + 120 * q^70 - 64 * q^72 + 200 * q^73 - 20 * q^75 - 64 * q^76 + 40 * q^78 - 112 * q^79 + 28 * q^81 + 192 * q^82 + 104 * q^84 - 120 * q^85 + 240 * q^87 - 96 * q^88 - 80 * q^90 - 80 * q^91 + 32 * q^93 - 120 * q^94 - 16 * q^96 + 296 * q^97 - 192 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 4x^{2} + 9$$ :

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1
 −1.58114 + 0.707107i 1.58114 + 0.707107i −1.58114 − 0.707107i 1.58114 − 0.707107i
1.41421i −0.581139 2.94317i −2.00000 2.23607i −4.16228 + 0.821854i 11.4868 2.82843i −8.32456 + 3.42079i 3.16228
11.2 1.41421i 2.58114 + 1.52896i −2.00000 2.23607i 2.16228 3.65028i −7.48683 2.82843i 4.32456 + 7.89292i −3.16228
11.3 1.41421i −0.581139 + 2.94317i −2.00000 2.23607i −4.16228 0.821854i 11.4868 2.82843i −8.32456 3.42079i 3.16228
11.4 1.41421i 2.58114 1.52896i −2.00000 2.23607i 2.16228 + 3.65028i −7.48683 2.82843i 4.32456 7.89292i −3.16228
 $$n$$: 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 30.3.d.a 4
3.b odd 2 1 inner 30.3.d.a 4
4.b odd 2 1 240.3.l.c 4
5.b even 2 1 150.3.d.c 4
5.c odd 4 2 150.3.b.b 8
8.b even 2 1 960.3.l.e 4
8.d odd 2 1 960.3.l.f 4
9.c even 3 2 810.3.h.a 8
9.d odd 6 2 810.3.h.a 8
12.b even 2 1 240.3.l.c 4
15.d odd 2 1 150.3.d.c 4
15.e even 4 2 150.3.b.b 8
20.d odd 2 1 1200.3.l.u 4
20.e even 4 2 1200.3.c.k 8
24.f even 2 1 960.3.l.f 4
24.h odd 2 1 960.3.l.e 4
60.h even 2 1 1200.3.l.u 4
60.l odd 4 2 1200.3.c.k 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.3.d.a 4 1.a even 1 1 trivial
30.3.d.a 4 3.b odd 2 1 inner
150.3.b.b 8 5.c odd 4 2
150.3.b.b 8 15.e even 4 2
150.3.d.c 4 5.b even 2 1
150.3.d.c 4 15.d odd 2 1
240.3.l.c 4 4.b odd 2 1
240.3.l.c 4 12.b even 2 1
810.3.h.a 8 9.c even 3 2
810.3.h.a 8 9.d odd 6 2
960.3.l.e 4 8.b even 2 1
960.3.l.e 4 24.h odd 2 1
960.3.l.f 4 8.d odd 2 1
960.3.l.f 4 24.f even 2 1
1200.3.c.k 8 20.e even 4 2
1200.3.c.k 8 60.l odd 4 2
1200.3.l.u 4 20.d odd 2 1
1200.3.l.u 4 60.h even 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 2)^{2}$$
$3$ $$T^{4} - 4 T^{3} + 12 T^{2} - 36 T + 81$$
$5$ $$(T^{2} + 5)^{2}$$
$7$ $$(T^{2} - 4 T - 86)^{2}$$
$11$ $$(T^{2} + 72)^{2}$$
$13$ $$(T + 10)^{4}$$
$17$ $$T^{4} + 936 T^{2} + 11664$$
$19$ $$(T^{2} - 16 T - 296)^{2}$$
$23$ $$T^{4} + 396 T^{2} + 26244$$
$29$ $$(T^{2} + 720)^{2}$$
$31$ $$(T - 8)^{4}$$
$37$ $$(T^{2} + 44 T - 956)^{2}$$
$41$ $$T^{4} + 2664 T^{2} + 944784$$
$43$ $$(T^{2} - 28 T - 614)^{2}$$
$47$ $$T^{4} + 9900 T^{2} + \cdots + 16402500$$
$53$ $$T^{4} + 936 T^{2} + 11664$$
$59$ $$T^{4} + 6624 T^{2} + \cdots + 3504384$$
$61$ $$(T^{2} + 32 T - 1184)^{2}$$
$67$ $$(T^{2} + 164 T + 5914)^{2}$$
$71$ $$T^{4} + 5040 T^{2} + \cdots + 1166400$$
$73$ $$(T^{2} - 100 T + 1060)^{2}$$
$79$ $$(T^{2} + 56 T + 424)^{2}$$
$83$ $$T^{4} + 684T^{2} + 324$$
$89$ $$T^{4} + 3744 T^{2} + 186624$$
$97$ $$(T^{2} - 148 T + 4036)^{2}$$