Properties

 Label 156.3.g.b Level $156$ Weight $3$ Character orbit 156.g Analytic conductor $4.251$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$156 = 2^{2} \cdot 3 \cdot 13$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 156.g (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$4.25069212402$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{11})$$ Defining polynomial: $$x^{4} - 5x^{2} + 9$$ x^4 - 5*x^2 + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + (\beta_{3} - 2) q^{3} + (\beta_{2} - 2 \beta_1) q^{5} - 7 \beta_{2} q^{7} + ( - 5 \beta_{3} + 1) q^{9}+O(q^{10})$$ q + (b3 - 2) * q^3 + (b2 - 2*b1) * q^5 - 7*b2 * q^7 + (-5*b3 + 1) * q^9 $$q + (\beta_{3} - 2) q^{3} + (\beta_{2} - 2 \beta_1) q^{5} - 7 \beta_{2} q^{7} + ( - 5 \beta_{3} + 1) q^{9} + ( - 6 \beta_{2} + 12 \beta_1) q^{11} - 13 \beta_{2} q^{13} + ( - 8 \beta_{2} + 5 \beta_1) q^{15} + (14 \beta_{3} + 7) q^{17} - 30 \beta_{2} q^{19} + (14 \beta_{2} + 7 \beta_1) q^{21} + ( - 16 \beta_{3} - 8) q^{23} - 14 q^{25} + (16 \beta_{3} + 13) q^{27} + ( - 20 \beta_{3} - 10) q^{29} - 10 \beta_{2} q^{31} + (48 \beta_{2} - 30 \beta_1) q^{33} + (14 \beta_{3} + 7) q^{35} + 13 \beta_{2} q^{37} + (26 \beta_{2} + 13 \beta_1) q^{39} + (14 \beta_{2} - 28 \beta_1) q^{41} + 55 q^{43} + (31 \beta_{2} - 7 \beta_1) q^{45} + (15 \beta_{2} - 30 \beta_1) q^{47} + ( - 35 \beta_{3} - 56) q^{51} + (24 \beta_{3} + 12) q^{53} - 66 q^{55} + (60 \beta_{2} + 30 \beta_1) q^{57} + ( - 28 \beta_{2} + 56 \beta_1) q^{59} + 14 q^{61} + ( - 7 \beta_{2} - 35 \beta_1) q^{63} + (26 \beta_{3} + 13) q^{65} - 16 \beta_{2} q^{67} + (40 \beta_{3} + 64) q^{69} + (9 \beta_{2} - 18 \beta_1) q^{71} + 4 \beta_{2} q^{73} + ( - 14 \beta_{3} + 28) q^{75} + ( - 84 \beta_{3} - 42) q^{77} + 54 q^{79} + ( - 35 \beta_{3} - 74) q^{81} + (10 \beta_{2} - 20 \beta_1) q^{83} - 77 \beta_{2} q^{85} + (50 \beta_{3} + 80) q^{87} + ( - 6 \beta_{2} + 12 \beta_1) q^{89} - 91 q^{91} + (20 \beta_{2} + 10 \beta_1) q^{93} + (60 \beta_{3} + 30) q^{95} + 78 \beta_{2} q^{97} + ( - 186 \beta_{2} + 42 \beta_1) q^{99}+O(q^{100})$$ q + (b3 - 2) * q^3 + (b2 - 2*b1) * q^5 - 7*b2 * q^7 + (-5*b3 + 1) * q^9 + (-6*b2 + 12*b1) * q^11 - 13*b2 * q^13 + (-8*b2 + 5*b1) * q^15 + (14*b3 + 7) * q^17 - 30*b2 * q^19 + (14*b2 + 7*b1) * q^21 + (-16*b3 - 8) * q^23 - 14 * q^25 + (16*b3 + 13) * q^27 + (-20*b3 - 10) * q^29 - 10*b2 * q^31 + (48*b2 - 30*b1) * q^33 + (14*b3 + 7) * q^35 + 13*b2 * q^37 + (26*b2 + 13*b1) * q^39 + (14*b2 - 28*b1) * q^41 + 55 * q^43 + (31*b2 - 7*b1) * q^45 + (15*b2 - 30*b1) * q^47 + (-35*b3 - 56) * q^51 + (24*b3 + 12) * q^53 - 66 * q^55 + (60*b2 + 30*b1) * q^57 + (-28*b2 + 56*b1) * q^59 + 14 * q^61 + (-7*b2 - 35*b1) * q^63 + (26*b3 + 13) * q^65 - 16*b2 * q^67 + (40*b3 + 64) * q^69 + (9*b2 - 18*b1) * q^71 + 4*b2 * q^73 + (-14*b3 + 28) * q^75 + (-84*b3 - 42) * q^77 + 54 * q^79 + (-35*b3 - 74) * q^81 + (10*b2 - 20*b1) * q^83 - 77*b2 * q^85 + (50*b3 + 80) * q^87 + (-6*b2 + 12*b1) * q^89 - 91 * q^91 + (20*b2 + 10*b1) * q^93 + (60*b3 + 30) * q^95 + 78*b2 * q^97 + (-186*b2 + 42*b1) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 10 q^{3} + 14 q^{9}+O(q^{10})$$ 4 * q - 10 * q^3 + 14 * q^9 $$4 q - 10 q^{3} + 14 q^{9} - 56 q^{25} + 20 q^{27} + 220 q^{43} - 154 q^{51} - 264 q^{55} + 56 q^{61} + 176 q^{69} + 140 q^{75} + 216 q^{79} - 226 q^{81} + 220 q^{87} - 364 q^{91}+O(q^{100})$$ 4 * q - 10 * q^3 + 14 * q^9 - 56 * q^25 + 20 * q^27 + 220 * q^43 - 154 * q^51 - 264 * q^55 + 56 * q^61 + 176 * q^69 + 140 * q^75 + 216 * q^79 - 226 * q^81 + 220 * q^87 - 364 * q^91

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5x^{2} + 9$$ :

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

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/156\mathbb{Z}\right)^\times$$.

 $$n$$ $$53$$ $$79$$ $$145$$ $$\chi(n)$$ $$-1$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
77.1
 1.65831 − 0.500000i −1.65831 + 0.500000i 1.65831 + 0.500000i −1.65831 − 0.500000i
0 −2.50000 1.65831i 0 −3.31662 0 7.00000i 0 3.50000 + 8.29156i 0
77.2 0 −2.50000 1.65831i 0 3.31662 0 7.00000i 0 3.50000 + 8.29156i 0
77.3 0 −2.50000 + 1.65831i 0 −3.31662 0 7.00000i 0 3.50000 8.29156i 0
77.4 0 −2.50000 + 1.65831i 0 3.31662 0 7.00000i 0 3.50000 8.29156i 0
 $$n$$: 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
13.b even 2 1 inner
39.d odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 156.3.g.b 4
3.b odd 2 1 inner 156.3.g.b 4
4.b odd 2 1 624.3.l.h 4
12.b even 2 1 624.3.l.h 4
13.b even 2 1 inner 156.3.g.b 4
39.d odd 2 1 inner 156.3.g.b 4
52.b odd 2 1 624.3.l.h 4
156.h even 2 1 624.3.l.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
156.3.g.b 4 1.a even 1 1 trivial
156.3.g.b 4 3.b odd 2 1 inner
156.3.g.b 4 13.b even 2 1 inner
156.3.g.b 4 39.d odd 2 1 inner
624.3.l.h 4 4.b odd 2 1
624.3.l.h 4 12.b even 2 1
624.3.l.h 4 52.b odd 2 1
624.3.l.h 4 156.h even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{2} - 11$$ acting on $$S_{3}^{\mathrm{new}}(156, [\chi])$$.

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$(T^{2} + 5 T + 9)^{2}$$
$5$ $$(T^{2} - 11)^{2}$$
$7$ $$(T^{2} + 49)^{2}$$
$11$ $$(T^{2} - 396)^{2}$$
$13$ $$(T^{2} + 169)^{2}$$
$17$ $$(T^{2} + 539)^{2}$$
$19$ $$(T^{2} + 900)^{2}$$
$23$ $$(T^{2} + 704)^{2}$$
$29$ $$(T^{2} + 1100)^{2}$$
$31$ $$(T^{2} + 100)^{2}$$
$37$ $$(T^{2} + 169)^{2}$$
$41$ $$(T^{2} - 2156)^{2}$$
$43$ $$(T - 55)^{4}$$
$47$ $$(T^{2} - 2475)^{2}$$
$53$ $$(T^{2} + 1584)^{2}$$
$59$ $$(T^{2} - 8624)^{2}$$
$61$ $$(T - 14)^{4}$$
$67$ $$(T^{2} + 256)^{2}$$
$71$ $$(T^{2} - 891)^{2}$$
$73$ $$(T^{2} + 16)^{2}$$
$79$ $$(T - 54)^{4}$$
$83$ $$(T^{2} - 1100)^{2}$$
$89$ $$(T^{2} - 396)^{2}$$
$97$ $$(T^{2} + 6084)^{2}$$