# Properties

 Label 361.3.d.b Level $361$ Weight $3$ Character orbit 361.d Analytic conductor $9.837$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$361 = 19^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 361.d (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.83653754341$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-13})$$ Defining polynomial: $$x^{4} - 13 x^{2} + 169$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 19) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} -\beta_{1} q^{3} + 9 \beta_{2} q^{4} + ( -4 + 4 \beta_{2} ) q^{5} -13 \beta_{2} q^{6} -5 q^{7} + 5 \beta_{3} q^{8} + 4 \beta_{2} q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} -\beta_{1} q^{3} + 9 \beta_{2} q^{4} + ( -4 + 4 \beta_{2} ) q^{5} -13 \beta_{2} q^{6} -5 q^{7} + 5 \beta_{3} q^{8} + 4 \beta_{2} q^{9} + ( -4 \beta_{1} + 4 \beta_{3} ) q^{10} -10 q^{11} -9 \beta_{3} q^{12} + ( -\beta_{1} + \beta_{3} ) q^{13} -5 \beta_{1} q^{14} + ( 4 \beta_{1} - 4 \beta_{3} ) q^{15} + ( -29 + 29 \beta_{2} ) q^{16} + ( -15 + 15 \beta_{2} ) q^{17} + 4 \beta_{3} q^{18} -36 q^{20} + 5 \beta_{1} q^{21} -10 \beta_{1} q^{22} -35 \beta_{2} q^{23} + ( 65 - 65 \beta_{2} ) q^{24} + 9 \beta_{2} q^{25} -13 q^{26} + 5 \beta_{3} q^{27} -45 \beta_{2} q^{28} + ( -5 \beta_{1} + 5 \beta_{3} ) q^{29} + 52 q^{30} + 10 \beta_{3} q^{31} + ( -9 \beta_{1} + 9 \beta_{3} ) q^{32} + 10 \beta_{1} q^{33} + ( -15 \beta_{1} + 15 \beta_{3} ) q^{34} + ( 20 - 20 \beta_{2} ) q^{35} + ( -36 + 36 \beta_{2} ) q^{36} + 6 \beta_{3} q^{37} + 13 q^{39} -20 \beta_{1} q^{40} + 10 \beta_{1} q^{41} + 65 \beta_{2} q^{42} + ( 20 - 20 \beta_{2} ) q^{43} -90 \beta_{2} q^{44} -16 q^{45} -35 \beta_{3} q^{46} -10 \beta_{2} q^{47} + ( 29 \beta_{1} - 29 \beta_{3} ) q^{48} -24 q^{49} + 9 \beta_{3} q^{50} + ( 15 \beta_{1} - 15 \beta_{3} ) q^{51} -9 \beta_{1} q^{52} + ( 21 \beta_{1} - 21 \beta_{3} ) q^{53} + ( -65 + 65 \beta_{2} ) q^{54} + ( 40 - 40 \beta_{2} ) q^{55} -25 \beta_{3} q^{56} -65 q^{58} + 5 \beta_{1} q^{59} + 36 \beta_{1} q^{60} + 40 \beta_{2} q^{61} + ( -130 + 130 \beta_{2} ) q^{62} -20 \beta_{2} q^{63} - q^{64} -4 \beta_{3} q^{65} + 130 \beta_{2} q^{66} + ( -11 \beta_{1} + 11 \beta_{3} ) q^{67} -135 q^{68} + 35 \beta_{3} q^{69} + ( 20 \beta_{1} - 20 \beta_{3} ) q^{70} + 30 \beta_{1} q^{71} + ( -20 \beta_{1} + 20 \beta_{3} ) q^{72} + ( -105 + 105 \beta_{2} ) q^{73} + ( -78 + 78 \beta_{2} ) q^{74} -9 \beta_{3} q^{75} + 50 q^{77} + 13 \beta_{1} q^{78} -10 \beta_{1} q^{79} -116 \beta_{2} q^{80} + ( 101 - 101 \beta_{2} ) q^{81} + 130 \beta_{2} q^{82} -40 q^{83} + 45 \beta_{3} q^{84} -60 \beta_{2} q^{85} + ( 20 \beta_{1} - 20 \beta_{3} ) q^{86} + 65 q^{87} -50 \beta_{3} q^{88} -16 \beta_{1} q^{90} + ( 5 \beta_{1} - 5 \beta_{3} ) q^{91} + ( 315 - 315 \beta_{2} ) q^{92} + ( 130 - 130 \beta_{2} ) q^{93} -10 \beta_{3} q^{94} + 117 q^{96} + 34 \beta_{1} q^{97} -24 \beta_{1} q^{98} -40 \beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 18q^{4} - 8q^{5} - 26q^{6} - 20q^{7} + 8q^{9} + O(q^{10})$$ $$4q + 18q^{4} - 8q^{5} - 26q^{6} - 20q^{7} + 8q^{9} - 40q^{11} - 58q^{16} - 30q^{17} - 144q^{20} - 70q^{23} + 130q^{24} + 18q^{25} - 52q^{26} - 90q^{28} + 208q^{30} + 40q^{35} - 72q^{36} + 52q^{39} + 130q^{42} + 40q^{43} - 180q^{44} - 64q^{45} - 20q^{47} - 96q^{49} - 130q^{54} + 80q^{55} - 260q^{58} + 80q^{61} - 260q^{62} - 40q^{63} - 4q^{64} + 260q^{66} - 540q^{68} - 210q^{73} - 156q^{74} + 200q^{77} - 232q^{80} + 202q^{81} + 260q^{82} - 160q^{83} - 120q^{85} + 260q^{87} + 630q^{92} + 260q^{93} + 468q^{96} - 80q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 13 x^{2} + 169$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/13$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/13$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$13 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$13 \beta_{3}$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/361\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$\beta_{2}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
69.1
 −3.12250 + 1.80278i 3.12250 − 1.80278i −3.12250 − 1.80278i 3.12250 + 1.80278i
−3.12250 + 1.80278i 3.12250 1.80278i 4.50000 7.79423i −2.00000 3.46410i −6.50000 + 11.2583i −5.00000 18.0278i 2.00000 3.46410i 12.4900 + 7.21110i
69.2 3.12250 1.80278i −3.12250 + 1.80278i 4.50000 7.79423i −2.00000 3.46410i −6.50000 + 11.2583i −5.00000 18.0278i 2.00000 3.46410i −12.4900 7.21110i
293.1 −3.12250 1.80278i 3.12250 + 1.80278i 4.50000 + 7.79423i −2.00000 + 3.46410i −6.50000 11.2583i −5.00000 18.0278i 2.00000 + 3.46410i 12.4900 7.21110i
293.2 3.12250 + 1.80278i −3.12250 1.80278i 4.50000 + 7.79423i −2.00000 + 3.46410i −6.50000 11.2583i −5.00000 18.0278i 2.00000 + 3.46410i −12.4900 + 7.21110i
 $$n$$: 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
19.b odd 2 1 inner
19.c even 3 1 inner
19.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 361.3.d.b 4
19.b odd 2 1 inner 361.3.d.b 4
19.c even 3 1 19.3.b.b 2
19.c even 3 1 inner 361.3.d.b 4
19.d odd 6 1 19.3.b.b 2
19.d odd 6 1 inner 361.3.d.b 4
19.e even 9 6 361.3.f.d 12
19.f odd 18 6 361.3.f.d 12
57.f even 6 1 171.3.c.b 2
57.h odd 6 1 171.3.c.b 2
76.f even 6 1 304.3.e.d 2
76.g odd 6 1 304.3.e.d 2
95.h odd 6 1 475.3.c.b 2
95.i even 6 1 475.3.c.b 2
95.l even 12 2 475.3.d.b 4
95.m odd 12 2 475.3.d.b 4
152.k odd 6 1 1216.3.e.h 2
152.l odd 6 1 1216.3.e.g 2
152.o even 6 1 1216.3.e.h 2
152.p even 6 1 1216.3.e.g 2
228.m even 6 1 2736.3.o.d 2
228.n odd 6 1 2736.3.o.d 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.3.b.b 2 19.c even 3 1
19.3.b.b 2 19.d odd 6 1
171.3.c.b 2 57.f even 6 1
171.3.c.b 2 57.h odd 6 1
304.3.e.d 2 76.f even 6 1
304.3.e.d 2 76.g odd 6 1
361.3.d.b 4 1.a even 1 1 trivial
361.3.d.b 4 19.b odd 2 1 inner
361.3.d.b 4 19.c even 3 1 inner
361.3.d.b 4 19.d odd 6 1 inner
361.3.f.d 12 19.e even 9 6
361.3.f.d 12 19.f odd 18 6
475.3.c.b 2 95.h odd 6 1
475.3.c.b 2 95.i even 6 1
475.3.d.b 4 95.l even 12 2
475.3.d.b 4 95.m odd 12 2
1216.3.e.g 2 152.l odd 6 1
1216.3.e.g 2 152.p even 6 1
1216.3.e.h 2 152.k odd 6 1
1216.3.e.h 2 152.o even 6 1
2736.3.o.d 2 228.m even 6 1
2736.3.o.d 2 228.n odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} - 13 T_{2}^{2} + 169$$ acting on $$S_{3}^{\mathrm{new}}(361, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$169 - 13 T^{2} + T^{4}$$
$3$ $$169 - 13 T^{2} + T^{4}$$
$5$ $$( 16 + 4 T + T^{2} )^{2}$$
$7$ $$( 5 + T )^{4}$$
$11$ $$( 10 + T )^{4}$$
$13$ $$169 - 13 T^{2} + T^{4}$$
$17$ $$( 225 + 15 T + T^{2} )^{2}$$
$19$ $$T^{4}$$
$23$ $$( 1225 + 35 T + T^{2} )^{2}$$
$29$ $$105625 - 325 T^{2} + T^{4}$$
$31$ $$( 1300 + T^{2} )^{2}$$
$37$ $$( 468 + T^{2} )^{2}$$
$41$ $$1690000 - 1300 T^{2} + T^{4}$$
$43$ $$( 400 - 20 T + T^{2} )^{2}$$
$47$ $$( 100 + 10 T + T^{2} )^{2}$$
$53$ $$32867289 - 5733 T^{2} + T^{4}$$
$59$ $$105625 - 325 T^{2} + T^{4}$$
$61$ $$( 1600 - 40 T + T^{2} )^{2}$$
$67$ $$2474329 - 1573 T^{2} + T^{4}$$
$71$ $$136890000 - 11700 T^{2} + T^{4}$$
$73$ $$( 11025 + 105 T + T^{2} )^{2}$$
$79$ $$1690000 - 1300 T^{2} + T^{4}$$
$83$ $$( 40 + T )^{4}$$
$89$ $$T^{4}$$
$97$ $$225840784 - 15028 T^{2} + T^{4}$$