# Properties

 Label 1900.2.s.b Level $1900$ Weight $2$ Character orbit 1900.s Analytic conductor $15.172$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1900 = 2^{2} \cdot 5^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1900.s (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$15.1715763840$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, \ldots, a_{9}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 380) 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.

 $$f(q)$$ $$=$$ $$q + \beta_1 q^{3} - 2 \beta_{3} q^{7} + \beta_{2} q^{9}+O(q^{10})$$ q + b1 * q^3 - 2*b3 * q^7 + b2 * q^9 $$q + \beta_1 q^{3} - 2 \beta_{3} q^{7} + \beta_{2} q^{9} - 3 q^{11} + (3 \beta_{3} - 3 \beta_1) q^{13} - \beta_1 q^{17} + ( - 3 \beta_{2} - 2) q^{19} + ( - 8 \beta_{2} + 8) q^{21} + (2 \beta_{3} - 2 \beta_1) q^{23} - 2 \beta_{3} q^{27} + \beta_{2} q^{29} - 5 q^{31} - 3 \beta_1 q^{33} - 2 \beta_{3} q^{37} - 12 q^{39} + (2 \beta_{2} - 2) q^{41} + ( - 3 \beta_{3} + 3 \beta_1) q^{47} - 9 q^{49} - 4 \beta_{2} q^{51} + ( - 3 \beta_{3} + 3 \beta_1) q^{53} + ( - 3 \beta_{3} - 2 \beta_1) q^{57} + (\beta_{2} - 1) q^{59} + 7 \beta_{2} q^{61} + ( - 2 \beta_{3} + 2 \beta_1) q^{63} + (7 \beta_{3} - 7 \beta_1) q^{67} - 8 q^{69} + (15 \beta_{2} - 15) q^{71} + 6 \beta_1 q^{73} + 6 \beta_{3} q^{77} + (\beta_{2} - 1) q^{79} + ( - 11 \beta_{2} + 11) q^{81} - 8 \beta_{3} q^{83} + \beta_{3} q^{87} + 17 \beta_{2} q^{89} + 24 \beta_{2} q^{91} - 5 \beta_1 q^{93} - 6 \beta_1 q^{97} - 3 \beta_{2} q^{99}+O(q^{100})$$ q + b1 * q^3 - 2*b3 * q^7 + b2 * q^9 - 3 * q^11 + (3*b3 - 3*b1) * q^13 - b1 * q^17 + (-3*b2 - 2) * q^19 + (-8*b2 + 8) * q^21 + (2*b3 - 2*b1) * q^23 - 2*b3 * q^27 + b2 * q^29 - 5 * q^31 - 3*b1 * q^33 - 2*b3 * q^37 - 12 * q^39 + (2*b2 - 2) * q^41 + (-3*b3 + 3*b1) * q^47 - 9 * q^49 - 4*b2 * q^51 + (-3*b3 + 3*b1) * q^53 + (-3*b3 - 2*b1) * q^57 + (b2 - 1) * q^59 + 7*b2 * q^61 + (-2*b3 + 2*b1) * q^63 + (7*b3 - 7*b1) * q^67 - 8 * q^69 + (15*b2 - 15) * q^71 + 6*b1 * q^73 + 6*b3 * q^77 + (b2 - 1) * q^79 + (-11*b2 + 11) * q^81 - 8*b3 * q^83 + b3 * q^87 + 17*b2 * q^89 + 24*b2 * q^91 - 5*b1 * q^93 - 6*b1 * q^97 - 3*b2 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{9}+O(q^{10})$$ 4 * q + 2 * q^9 $$4 q + 2 q^{9} - 12 q^{11} - 14 q^{19} + 16 q^{21} + 2 q^{29} - 20 q^{31} - 48 q^{39} - 4 q^{41} - 36 q^{49} - 8 q^{51} - 2 q^{59} + 14 q^{61} - 32 q^{69} - 30 q^{71} - 2 q^{79} + 22 q^{81} + 34 q^{89} + 48 q^{91} - 6 q^{99}+O(q^{100})$$ 4 * q + 2 * q^9 - 12 * q^11 - 14 * q^19 + 16 * q^21 + 2 * q^29 - 20 * q^31 - 48 * q^39 - 4 * q^41 - 36 * q^49 - 8 * q^51 - 2 * q^59 + 14 * q^61 - 32 * q^69 - 30 * q^71 - 2 * q^79 + 22 * q^81 + 34 * q^89 + 48 * q^91 - 6 * q^99

Basis of coefficient ring

 $$\beta_{1}$$ $$=$$ $$2\zeta_{12}$$ 2*v $$\beta_{2}$$ $$=$$ $$\zeta_{12}^{2}$$ v^2 $$\beta_{3}$$ $$=$$ $$2\zeta_{12}^{3}$$ 2*v^3
 $$\zeta_{12}$$ $$=$$ $$( \beta_1 ) / 2$$ (b1) / 2 $$\zeta_{12}^{2}$$ $$=$$ $$\beta_{2}$$ b2 $$\zeta_{12}^{3}$$ $$=$$ $$( \beta_{3} ) / 2$$ (b3) / 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1900\mathbb{Z}\right)^\times$$.

 $$n$$ $$77$$ $$401$$ $$951$$ $$\chi(n)$$ $$-1$$ $$-\beta_{2}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
49.1
 −0.866025 − 0.500000i 0.866025 + 0.500000i −0.866025 + 0.500000i 0.866025 − 0.500000i
0 −1.73205 1.00000i 0 0 0 4.00000i 0 0.500000 + 0.866025i 0
49.2 0 1.73205 + 1.00000i 0 0 0 4.00000i 0 0.500000 + 0.866025i 0
349.1 0 −1.73205 + 1.00000i 0 0 0 4.00000i 0 0.500000 0.866025i 0
349.2 0 1.73205 1.00000i 0 0 0 4.00000i 0 0.500000 0.866025i 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
5.b even 2 1 inner
19.c even 3 1 inner
95.i even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1900.2.s.b 4
5.b even 2 1 inner 1900.2.s.b 4
5.c odd 4 1 380.2.i.a 2
5.c odd 4 1 1900.2.i.b 2
15.e even 4 1 3420.2.t.b 2
19.c even 3 1 inner 1900.2.s.b 4
20.e even 4 1 1520.2.q.g 2
95.i even 6 1 inner 1900.2.s.b 4
95.l even 12 1 7220.2.a.a 1
95.m odd 12 1 380.2.i.a 2
95.m odd 12 1 1900.2.i.b 2
95.m odd 12 1 7220.2.a.e 1
285.v even 12 1 3420.2.t.b 2
380.v even 12 1 1520.2.q.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
380.2.i.a 2 5.c odd 4 1
380.2.i.a 2 95.m odd 12 1
1520.2.q.g 2 20.e even 4 1
1520.2.q.g 2 380.v even 12 1
1900.2.i.b 2 5.c odd 4 1
1900.2.i.b 2 95.m odd 12 1
1900.2.s.b 4 1.a even 1 1 trivial
1900.2.s.b 4 5.b even 2 1 inner
1900.2.s.b 4 19.c even 3 1 inner
1900.2.s.b 4 95.i even 6 1 inner
3420.2.t.b 2 15.e even 4 1
3420.2.t.b 2 285.v even 12 1
7220.2.a.a 1 95.l even 12 1
7220.2.a.e 1 95.m odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{4} - 4T_{3}^{2} + 16$$ acting on $$S_{2}^{\mathrm{new}}(1900, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} - 4T^{2} + 16$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 16)^{2}$$
$11$ $$(T + 3)^{4}$$
$13$ $$T^{4} - 36T^{2} + 1296$$
$17$ $$T^{4} - 4T^{2} + 16$$
$19$ $$(T^{2} + 7 T + 19)^{2}$$
$23$ $$T^{4} - 16T^{2} + 256$$
$29$ $$(T^{2} - T + 1)^{2}$$
$31$ $$(T + 5)^{4}$$
$37$ $$(T^{2} + 16)^{2}$$
$41$ $$(T^{2} + 2 T + 4)^{2}$$
$43$ $$T^{4}$$
$47$ $$T^{4} - 36T^{2} + 1296$$
$53$ $$T^{4} - 36T^{2} + 1296$$
$59$ $$(T^{2} + T + 1)^{2}$$
$61$ $$(T^{2} - 7 T + 49)^{2}$$
$67$ $$T^{4} - 196 T^{2} + 38416$$
$71$ $$(T^{2} + 15 T + 225)^{2}$$
$73$ $$T^{4} - 144 T^{2} + 20736$$
$79$ $$(T^{2} + T + 1)^{2}$$
$83$ $$(T^{2} + 256)^{2}$$
$89$ $$(T^{2} - 17 T + 289)^{2}$$
$97$ $$T^{4} - 144 T^{2} + 20736$$