Properties

 Label 1850.2.d.e Level $1850$ Weight $2$ Character orbit 1850.d 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.d (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$14.7723243739$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{21})$$ Defining polynomial: $$x^{4} + 11x^{2} + 25$$ x^4 + 11*x^2 + 25 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 74) 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.

 $$f(q)$$ $$=$$ $$q + \beta_{2} q^{2} + \beta_{3} q^{3} - q^{4} + (\beta_{2} + \beta_1) q^{6} + 2 q^{7} - \beta_{2} q^{8} + (\beta_{3} + 2) q^{9}+O(q^{10})$$ q + b2 * q^2 + b3 * q^3 - q^4 + (b2 + b1) * q^6 + 2 * q^7 - b2 * q^8 + (b3 + 2) * q^9 $$q + \beta_{2} q^{2} + \beta_{3} q^{3} - q^{4} + (\beta_{2} + \beta_1) q^{6} + 2 q^{7} - \beta_{2} q^{8} + (\beta_{3} + 2) q^{9} + (\beta_{3} + 1) q^{11} - \beta_{3} q^{12} + (\beta_{2} - \beta_1) q^{13} + 2 \beta_{2} q^{14} + q^{16} + ( - 2 \beta_{2} + 2 \beta_1) q^{17} + (3 \beta_{2} + \beta_1) q^{18} + ( - 4 \beta_{2} - 2 \beta_1) q^{19} + 2 \beta_{3} q^{21} + (2 \beta_{2} + \beta_1) q^{22} + (\beta_{2} - \beta_1) q^{23} + ( - \beta_{2} - \beta_1) q^{24} + (\beta_{3} - 2) q^{26} + 5 q^{27} - 2 q^{28} + (\beta_{2} - \beta_1) q^{29} + 3 \beta_1 q^{31} + \beta_{2} q^{32} + (2 \beta_{3} + 5) q^{33} + ( - 2 \beta_{3} + 4) q^{34} + ( - \beta_{3} - 2) q^{36} + ( - \beta_{2} - 2 \beta_1 - 4) q^{37} + (2 \beta_{3} + 2) q^{38} + ( - 4 \beta_{2} + \beta_1) q^{39} + ( - \beta_{3} + 8) q^{41} + (2 \beta_{2} + 2 \beta_1) q^{42} + 6 \beta_{2} q^{43} + ( - \beta_{3} - 1) q^{44} + (\beta_{3} - 2) q^{46} + ( - 2 \beta_{3} + 4) q^{47} + \beta_{3} q^{48} - 3 q^{49} + (8 \beta_{2} - 2 \beta_1) q^{51} + ( - \beta_{2} + \beta_1) q^{52} + ( - 2 \beta_{3} - 2) q^{53} + 5 \beta_{2} q^{54} - 2 \beta_{2} q^{56} + ( - 14 \beta_{2} - 4 \beta_1) q^{57} + (\beta_{3} - 2) q^{58} + ( - 4 \beta_{2} - 2 \beta_1) q^{59} + (10 \beta_{2} - \beta_1) q^{61} + ( - 3 \beta_{3} + 3) q^{62} + (2 \beta_{3} + 4) q^{63} - q^{64} + (7 \beta_{2} + 2 \beta_1) q^{66} + (3 \beta_{3} - 1) q^{67} + (2 \beta_{2} - 2 \beta_1) q^{68} + ( - 4 \beta_{2} + \beta_1) q^{69} + (4 \beta_{3} - 2) q^{71} + ( - 3 \beta_{2} - \beta_1) q^{72} + ( - 3 \beta_{3} - 1) q^{73} + (2 \beta_{3} - 4 \beta_{2} - 1) q^{74} + (4 \beta_{2} + 2 \beta_1) q^{76} + (2 \beta_{3} + 2) q^{77} + ( - \beta_{3} + 5) q^{78} + (11 \beta_{2} + \beta_1) q^{79} + (2 \beta_{3} - 6) q^{81} + (7 \beta_{2} - \beta_1) q^{82} + (4 \beta_{3} - 8) q^{83} - 2 \beta_{3} q^{84} - 6 q^{86} + ( - 4 \beta_{2} + \beta_1) q^{87} + ( - 2 \beta_{2} - \beta_1) q^{88} + 6 \beta_{2} q^{89} + (2 \beta_{2} - 2 \beta_1) q^{91} + ( - \beta_{2} + \beta_1) q^{92} + 15 \beta_{2} q^{93} + (2 \beta_{2} - 2 \beta_1) q^{94} + (\beta_{2} + \beta_1) q^{96} + (8 \beta_{2} - 2 \beta_1) q^{97} - 3 \beta_{2} q^{98} + (4 \beta_{3} + 7) q^{99}+O(q^{100})$$ q + b2 * q^2 + b3 * q^3 - q^4 + (b2 + b1) * q^6 + 2 * q^7 - b2 * q^8 + (b3 + 2) * q^9 + (b3 + 1) * q^11 - b3 * q^12 + (b2 - b1) * q^13 + 2*b2 * q^14 + q^16 + (-2*b2 + 2*b1) * q^17 + (3*b2 + b1) * q^18 + (-4*b2 - 2*b1) * q^19 + 2*b3 * q^21 + (2*b2 + b1) * q^22 + (b2 - b1) * q^23 + (-b2 - b1) * q^24 + (b3 - 2) * q^26 + 5 * q^27 - 2 * q^28 + (b2 - b1) * q^29 + 3*b1 * q^31 + b2 * q^32 + (2*b3 + 5) * q^33 + (-2*b3 + 4) * q^34 + (-b3 - 2) * q^36 + (-b2 - 2*b1 - 4) * q^37 + (2*b3 + 2) * q^38 + (-4*b2 + b1) * q^39 + (-b3 + 8) * q^41 + (2*b2 + 2*b1) * q^42 + 6*b2 * q^43 + (-b3 - 1) * q^44 + (b3 - 2) * q^46 + (-2*b3 + 4) * q^47 + b3 * q^48 - 3 * q^49 + (8*b2 - 2*b1) * q^51 + (-b2 + b1) * q^52 + (-2*b3 - 2) * q^53 + 5*b2 * q^54 - 2*b2 * q^56 + (-14*b2 - 4*b1) * q^57 + (b3 - 2) * q^58 + (-4*b2 - 2*b1) * q^59 + (10*b2 - b1) * q^61 + (-3*b3 + 3) * q^62 + (2*b3 + 4) * q^63 - q^64 + (7*b2 + 2*b1) * q^66 + (3*b3 - 1) * q^67 + (2*b2 - 2*b1) * q^68 + (-4*b2 + b1) * q^69 + (4*b3 - 2) * q^71 + (-3*b2 - b1) * q^72 + (-3*b3 - 1) * q^73 + (2*b3 - 4*b2 - 1) * q^74 + (4*b2 + 2*b1) * q^76 + (2*b3 + 2) * q^77 + (-b3 + 5) * q^78 + (11*b2 + b1) * q^79 + (2*b3 - 6) * q^81 + (7*b2 - b1) * q^82 + (4*b3 - 8) * q^83 - 2*b3 * q^84 - 6 * q^86 + (-4*b2 + b1) * q^87 + (-2*b2 - b1) * q^88 + 6*b2 * q^89 + (2*b2 - 2*b1) * q^91 + (-b2 + b1) * q^92 + 15*b2 * q^93 + (2*b2 - 2*b1) * q^94 + (b2 + b1) * q^96 + (8*b2 - 2*b1) * q^97 - 3*b2 * q^98 + (4*b3 + 7) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{3} - 4 q^{4} + 8 q^{7} + 10 q^{9}+O(q^{10})$$ 4 * q + 2 * q^3 - 4 * q^4 + 8 * q^7 + 10 * q^9 $$4 q + 2 q^{3} - 4 q^{4} + 8 q^{7} + 10 q^{9} + 6 q^{11} - 2 q^{12} + 4 q^{16} + 4 q^{21} - 6 q^{26} + 20 q^{27} - 8 q^{28} + 24 q^{33} + 12 q^{34} - 10 q^{36} - 16 q^{37} + 12 q^{38} + 30 q^{41} - 6 q^{44} - 6 q^{46} + 12 q^{47} + 2 q^{48} - 12 q^{49} - 12 q^{53} - 6 q^{58} + 6 q^{62} + 20 q^{63} - 4 q^{64} + 2 q^{67} - 10 q^{73} + 12 q^{77} + 18 q^{78} - 20 q^{81} - 24 q^{83} - 4 q^{84} - 24 q^{86} + 36 q^{99}+O(q^{100})$$ 4 * q + 2 * q^3 - 4 * q^4 + 8 * q^7 + 10 * q^9 + 6 * q^11 - 2 * q^12 + 4 * q^16 + 4 * q^21 - 6 * q^26 + 20 * q^27 - 8 * q^28 + 24 * q^33 + 12 * q^34 - 10 * q^36 - 16 * q^37 + 12 * q^38 + 30 * q^41 - 6 * q^44 - 6 * q^46 + 12 * q^47 + 2 * q^48 - 12 * q^49 - 12 * q^53 - 6 * q^58 + 6 * q^62 + 20 * q^63 - 4 * q^64 + 2 * q^67 - 10 * q^73 + 12 * q^77 + 18 * q^78 - 20 * q^81 - 24 * q^83 - 4 * q^84 - 24 * q^86 + 36 * q^99

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 11x^{2} + 25$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( \nu^{3} + 6\nu ) / 5$$ (v^3 + 6*v) / 5 $$\beta_{3}$$ $$=$$ $$\nu^{2} + 6$$ v^2 + 6
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{3} - 6$$ b3 - 6 $$\nu^{3}$$ $$=$$ $$5\beta_{2} - 6\beta_1$$ 5*b2 - 6*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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1701.1
 2.79129i − 1.79129i − 2.79129i 1.79129i
1.00000i −1.79129 −1.00000 0 1.79129i 2.00000 1.00000i 0.208712 0
1701.2 1.00000i 2.79129 −1.00000 0 2.79129i 2.00000 1.00000i 4.79129 0
1701.3 1.00000i −1.79129 −1.00000 0 1.79129i 2.00000 1.00000i 0.208712 0
1701.4 1.00000i 2.79129 −1.00000 0 2.79129i 2.00000 1.00000i 4.79129 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
37.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.d.e 4
5.b even 2 1 74.2.b.a 4
5.c odd 4 1 1850.2.c.g 4
5.c odd 4 1 1850.2.c.h 4
15.d odd 2 1 666.2.c.b 4
20.d odd 2 1 592.2.g.c 4
37.b even 2 1 inner 1850.2.d.e 4
40.e odd 2 1 2368.2.g.h 4
40.f even 2 1 2368.2.g.j 4
60.h even 2 1 5328.2.h.m 4
185.d even 2 1 74.2.b.a 4
185.h odd 4 1 1850.2.c.g 4
185.h odd 4 1 1850.2.c.h 4
185.j odd 4 1 2738.2.a.h 2
185.j odd 4 1 2738.2.a.k 2
555.b odd 2 1 666.2.c.b 4
740.g odd 2 1 592.2.g.c 4
1480.h odd 2 1 2368.2.g.h 4
1480.j even 2 1 2368.2.g.j 4
2220.p even 2 1 5328.2.h.m 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
74.2.b.a 4 5.b even 2 1
74.2.b.a 4 185.d even 2 1
592.2.g.c 4 20.d odd 2 1
592.2.g.c 4 740.g odd 2 1
666.2.c.b 4 15.d odd 2 1
666.2.c.b 4 555.b odd 2 1
1850.2.c.g 4 5.c odd 4 1
1850.2.c.g 4 185.h odd 4 1
1850.2.c.h 4 5.c odd 4 1
1850.2.c.h 4 185.h odd 4 1
1850.2.d.e 4 1.a even 1 1 trivial
1850.2.d.e 4 37.b even 2 1 inner
2368.2.g.h 4 40.e odd 2 1
2368.2.g.h 4 1480.h odd 2 1
2368.2.g.j 4 40.f even 2 1
2368.2.g.j 4 1480.j even 2 1
2738.2.a.h 2 185.j odd 4 1
2738.2.a.k 2 185.j odd 4 1
5328.2.h.m 4 60.h even 2 1
5328.2.h.m 4 2220.p even 2 1

Hecke kernels

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

 $$T_{3}^{2} - T_{3} - 5$$ T3^2 - T3 - 5 $$T_{7} - 2$$ T7 - 2 $$T_{11}^{2} - 3T_{11} - 3$$ T11^2 - 3*T11 - 3

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T^{2} + 1)^{2}$$
$3$ $$(T^{2} - T - 5)^{2}$$
$5$ $$T^{4}$$
$7$ $$(T - 2)^{4}$$
$11$ $$(T^{2} - 3 T - 3)^{2}$$
$13$ $$T^{4} + 15T^{2} + 9$$
$17$ $$T^{4} + 60T^{2} + 144$$
$19$ $$T^{4} + 60T^{2} + 144$$
$23$ $$T^{4} + 15T^{2} + 9$$
$29$ $$T^{4} + 15T^{2} + 9$$
$31$ $$T^{4} + 99T^{2} + 2025$$
$37$ $$(T^{2} + 8 T + 37)^{2}$$
$41$ $$(T^{2} - 15 T + 51)^{2}$$
$43$ $$(T^{2} + 36)^{2}$$
$47$ $$(T^{2} - 6 T - 12)^{2}$$
$53$ $$(T^{2} + 6 T - 12)^{2}$$
$59$ $$T^{4} + 60T^{2} + 144$$
$61$ $$T^{4} + 231 T^{2} + 11025$$
$67$ $$(T^{2} - T - 47)^{2}$$
$71$ $$(T^{2} - 84)^{2}$$
$73$ $$(T^{2} + 5 T - 41)^{2}$$
$79$ $$T^{4} + 231 T^{2} + 11025$$
$83$ $$(T^{2} + 12 T - 48)^{2}$$
$89$ $$(T^{2} + 36)^{2}$$
$97$ $$T^{4} + 204T^{2} + 3600$$