Properties

 Label 1850.2.b.l Level $1850$ Weight $2$ Character orbit 1850.b Analytic conductor $14.772$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.b (of order $$2$$, degree $$1$$, not minimal)

Newform invariants

 Self dual: no Analytic conductor: $$14.7723243739$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ x^4 - x^2 + 1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: no (minimal twist has level 370) 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.

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

Basis of coefficient ring

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

Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1001$$ $$1777$$ $$\chi(n)$$ $$1$$ $$-1$$

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1
 0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i
1.00000i 0.732051i −1.00000 0 −0.732051 4.73205i 1.00000i 2.46410 0
149.2 1.00000i 2.73205i −1.00000 0 2.73205 1.26795i 1.00000i −4.46410 0
149.3 1.00000i 2.73205i −1.00000 0 2.73205 1.26795i 1.00000i −4.46410 0
149.4 1.00000i 0.732051i −1.00000 0 −0.732051 4.73205i 1.00000i 2.46410 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

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1850.2.b.l 4
5.b even 2 1 inner 1850.2.b.l 4
5.c odd 4 1 370.2.a.e 2
5.c odd 4 1 1850.2.a.x 2
15.e even 4 1 3330.2.a.bd 2
20.e even 4 1 2960.2.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.a.e 2 5.c odd 4 1
1850.2.a.x 2 5.c odd 4 1
1850.2.b.l 4 1.a even 1 1 trivial
1850.2.b.l 4 5.b even 2 1 inner
2960.2.a.q 2 20.e even 4 1
3330.2.a.bd 2 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1850, [\chi])$$:

 $$T_{3}^{4} + 8T_{3}^{2} + 4$$ T3^4 + 8*T3^2 + 4 $$T_{7}^{4} + 24T_{7}^{2} + 36$$ T7^4 + 24*T7^2 + 36 $$T_{13}^{4} + 32T_{13}^{2} + 64$$ T13^4 + 32*T13^2 + 64

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$T^{4} + 8T^{2} + 4$$
$5$ $$T^{4}$$
$7$ $$T^{4} + 24T^{2} + 36$$
$11$ $$(T^{2} + 4 T - 8)^{2}$$
$13$ $$T^{4} + 32T^{2} + 64$$
$17$ $$T^{4} + 32T^{2} + 64$$
$19$ $$(T^{2} + 2 T - 26)^{2}$$
$23$ $$(T^{2} + 64)^{2}$$
$29$ $$(T^{2} - 4 T - 44)^{2}$$
$31$ $$(T^{2} + 2 T - 2)^{2}$$
$37$ $$(T^{2} + 1)^{2}$$
$41$ $$(T + 2)^{4}$$
$43$ $$(T^{2} + 48)^{2}$$
$47$ $$T^{4} + 24T^{2} + 36$$
$53$ $$(T^{2} + 36)^{2}$$
$59$ $$(T^{2} - 10 T - 2)^{2}$$
$61$ $$(T^{2} - 4 T - 44)^{2}$$
$67$ $$T^{4} + 200T^{2} + 2500$$
$71$ $$(T^{2} + 8 T - 32)^{2}$$
$73$ $$T^{4} + 168T^{2} + 144$$
$79$ $$(T^{2} + 14 T + 46)^{2}$$
$83$ $$T^{4} + 104T^{2} + 2116$$
$89$ $$(T - 2)^{4}$$
$97$ $$(T^{2} + 4)^{2}$$