# Properties

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

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1850 = 2 \cdot 5^{2} \cdot 37$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1850.c (of order $$2$$, degree $$1$$, not 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: $$\Z[a_1, \ldots, a_{7}]$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + \beta_1 q^{3} + q^{4} + \beta_1 q^{6} + 2 \beta_{2} q^{7} + q^{8} + (\beta_{3} - 3) q^{9}+O(q^{10})$$ q + q^2 + b1 * q^3 + q^4 + b1 * q^6 + 2*b2 * q^7 + q^8 + (b3 - 3) * q^9 $$q + q^{2} + \beta_1 q^{3} + q^{4} + \beta_1 q^{6} + 2 \beta_{2} q^{7} + q^{8} + (\beta_{3} - 3) q^{9} + ( - \beta_{3} + 2) q^{11} + \beta_1 q^{12} + ( - \beta_{3} - 1) q^{13} + 2 \beta_{2} q^{14} + q^{16} + ( - 2 \beta_{3} - 2) q^{17} + (\beta_{3} - 3) q^{18} + ( - 2 \beta_{2} + 2 \beta_1) q^{19} + ( - 2 \beta_{3} + 2) q^{21} + ( - \beta_{3} + 2) q^{22} + ( - \beta_{3} - 1) q^{23} + \beta_1 q^{24} + ( - \beta_{3} - 1) q^{26} + 5 \beta_{2} q^{27} + 2 \beta_{2} q^{28} + (2 \beta_{2} + \beta_1) q^{29} + (3 \beta_{2} + 3 \beta_1) q^{31} + q^{32} + ( - 5 \beta_{2} + 2 \beta_1) q^{33} + ( - 2 \beta_{3} - 2) q^{34} + (\beta_{3} - 3) q^{36} + (2 \beta_{3} - 4 \beta_{2} - 1) q^{37} + ( - 2 \beta_{2} + 2 \beta_1) q^{38} + ( - 5 \beta_{2} - \beta_1) q^{39} + (\beta_{3} + 7) q^{41} + ( - 2 \beta_{3} + 2) q^{42} - 6 q^{43} + ( - \beta_{3} + 2) q^{44} + ( - \beta_{3} - 1) q^{46} + (4 \beta_{2} + 2 \beta_1) q^{47} + \beta_1 q^{48} + 3 q^{49} + ( - 10 \beta_{2} - 2 \beta_1) q^{51} + ( - \beta_{3} - 1) q^{52} + (2 \beta_{2} - 2 \beta_1) q^{53} + 5 \beta_{2} q^{54} + 2 \beta_{2} q^{56} + (4 \beta_{3} - 14) q^{57} + (2 \beta_{2} + \beta_1) q^{58} + ( - 2 \beta_{2} + 2 \beta_1) q^{59} + ( - 11 \beta_{2} - \beta_1) q^{61} + (3 \beta_{2} + 3 \beta_1) q^{62} + ( - 4 \beta_{2} + 2 \beta_1) q^{63} + q^{64} + ( - 5 \beta_{2} + 2 \beta_1) q^{66} + ( - \beta_{2} - 3 \beta_1) q^{67} + ( - 2 \beta_{3} - 2) q^{68} + ( - 5 \beta_{2} - \beta_1) q^{69} + ( - 4 \beta_{3} + 2) q^{71} + (\beta_{3} - 3) q^{72} + (\beta_{2} - 3 \beta_1) q^{73} + (2 \beta_{3} - 4 \beta_{2} - 1) q^{74} + ( - 2 \beta_{2} + 2 \beta_1) q^{76} + (2 \beta_{2} - 2 \beta_1) q^{77} + ( - 5 \beta_{2} - \beta_1) q^{78} + (10 \beta_{2} - \beta_1) q^{79} + ( - 2 \beta_{3} - 4) q^{81} + (\beta_{3} + 7) q^{82} + (8 \beta_{2} + 4 \beta_1) q^{83} + ( - 2 \beta_{3} + 2) q^{84} - 6 q^{86} + ( - \beta_{3} - 4) q^{87} + ( - \beta_{3} + 2) q^{88} + 6 \beta_{2} q^{89} + ( - 4 \beta_{2} - 2 \beta_1) q^{91} + ( - \beta_{3} - 1) q^{92} - 15 q^{93} + (4 \beta_{2} + 2 \beta_1) q^{94} + \beta_1 q^{96} + (2 \beta_{3} + 8) q^{97} + 3 q^{98} + (4 \beta_{3} - 11) q^{99}+O(q^{100})$$ q + q^2 + b1 * q^3 + q^4 + b1 * q^6 + 2*b2 * q^7 + q^8 + (b3 - 3) * q^9 + (-b3 + 2) * q^11 + b1 * q^12 + (-b3 - 1) * q^13 + 2*b2 * q^14 + q^16 + (-2*b3 - 2) * q^17 + (b3 - 3) * q^18 + (-2*b2 + 2*b1) * q^19 + (-2*b3 + 2) * q^21 + (-b3 + 2) * q^22 + (-b3 - 1) * q^23 + b1 * q^24 + (-b3 - 1) * q^26 + 5*b2 * q^27 + 2*b2 * q^28 + (2*b2 + b1) * q^29 + (3*b2 + 3*b1) * q^31 + q^32 + (-5*b2 + 2*b1) * q^33 + (-2*b3 - 2) * q^34 + (b3 - 3) * q^36 + (2*b3 - 4*b2 - 1) * q^37 + (-2*b2 + 2*b1) * q^38 + (-5*b2 - b1) * q^39 + (b3 + 7) * q^41 + (-2*b3 + 2) * q^42 - 6 * q^43 + (-b3 + 2) * q^44 + (-b3 - 1) * q^46 + (4*b2 + 2*b1) * q^47 + b1 * q^48 + 3 * q^49 + (-10*b2 - 2*b1) * q^51 + (-b3 - 1) * q^52 + (2*b2 - 2*b1) * q^53 + 5*b2 * q^54 + 2*b2 * q^56 + (4*b3 - 14) * q^57 + (2*b2 + b1) * q^58 + (-2*b2 + 2*b1) * q^59 + (-11*b2 - b1) * q^61 + (3*b2 + 3*b1) * q^62 + (-4*b2 + 2*b1) * q^63 + q^64 + (-5*b2 + 2*b1) * q^66 + (-b2 - 3*b1) * q^67 + (-2*b3 - 2) * q^68 + (-5*b2 - b1) * q^69 + (-4*b3 + 2) * q^71 + (b3 - 3) * q^72 + (b2 - 3*b1) * q^73 + (2*b3 - 4*b2 - 1) * q^74 + (-2*b2 + 2*b1) * q^76 + (2*b2 - 2*b1) * q^77 + (-5*b2 - b1) * q^78 + (10*b2 - b1) * q^79 + (-2*b3 - 4) * q^81 + (b3 + 7) * q^82 + (8*b2 + 4*b1) * q^83 + (-2*b3 + 2) * q^84 - 6 * q^86 + (-b3 - 4) * q^87 + (-b3 + 2) * q^88 + 6*b2 * q^89 + (-4*b2 - 2*b1) * q^91 + (-b3 - 1) * q^92 - 15 * q^93 + (4*b2 + 2*b1) * q^94 + b1 * q^96 + (2*b3 + 8) * q^97 + 3 * q^98 + (4*b3 - 11) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} - 10 q^{9}+O(q^{10})$$ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 - 10 * q^9 $$4 q + 4 q^{2} + 4 q^{4} + 4 q^{8} - 10 q^{9} + 6 q^{11} - 6 q^{13} + 4 q^{16} - 12 q^{17} - 10 q^{18} + 4 q^{21} + 6 q^{22} - 6 q^{23} - 6 q^{26} + 4 q^{32} - 12 q^{34} - 10 q^{36} + 30 q^{41} + 4 q^{42} - 24 q^{43} + 6 q^{44} - 6 q^{46} + 12 q^{49} - 6 q^{52} - 48 q^{57} + 4 q^{64} - 12 q^{68} - 10 q^{72} - 20 q^{81} + 30 q^{82} + 4 q^{84} - 24 q^{86} - 18 q^{87} + 6 q^{88} - 6 q^{92} - 60 q^{93} + 36 q^{97} + 12 q^{98} - 36 q^{99}+O(q^{100})$$ 4 * q + 4 * q^2 + 4 * q^4 + 4 * q^8 - 10 * q^9 + 6 * q^11 - 6 * q^13 + 4 * q^16 - 12 * q^17 - 10 * q^18 + 4 * q^21 + 6 * q^22 - 6 * q^23 - 6 * q^26 + 4 * q^32 - 12 * q^34 - 10 * q^36 + 30 * q^41 + 4 * q^42 - 24 * q^43 + 6 * q^44 - 6 * q^46 + 12 * q^49 - 6 * q^52 - 48 * q^57 + 4 * q^64 - 12 * q^68 - 10 * q^72 - 20 * q^81 + 30 * q^82 + 4 * q^84 - 24 * q^86 - 18 * q^87 + 6 * q^88 - 6 * q^92 - 60 * q^93 + 36 * q^97 + 12 * q^98 - 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1849.1
 − 2.79129i − 1.79129i 1.79129i 2.79129i
1.00000 2.79129i 1.00000 0 2.79129i 2.00000i 1.00000 −4.79129 0
1849.2 1.00000 1.79129i 1.00000 0 1.79129i 2.00000i 1.00000 −0.208712 0
1849.3 1.00000 1.79129i 1.00000 0 1.79129i 2.00000i 1.00000 −0.208712 0
1849.4 1.00000 2.79129i 1.00000 0 2.79129i 2.00000i 1.00000 −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
185.d 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.c.h 4
5.b even 2 1 1850.2.c.g 4
5.c odd 4 1 74.2.b.a 4
5.c odd 4 1 1850.2.d.e 4
15.e even 4 1 666.2.c.b 4
20.e even 4 1 592.2.g.c 4
37.b even 2 1 1850.2.c.g 4
40.i odd 4 1 2368.2.g.j 4
40.k even 4 1 2368.2.g.h 4
60.l odd 4 1 5328.2.h.m 4
185.d even 2 1 inner 1850.2.c.h 4
185.f even 4 1 2738.2.a.h 2
185.h odd 4 1 74.2.b.a 4
185.h odd 4 1 1850.2.d.e 4
185.k even 4 1 2738.2.a.k 2
555.n even 4 1 666.2.c.b 4
740.m even 4 1 592.2.g.c 4
1480.x odd 4 1 2368.2.g.j 4
1480.bh even 4 1 2368.2.g.h 4
2220.bf odd 4 1 5328.2.h.m 4

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

## Hecke kernels

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

 $$T_{3}^{4} + 11T_{3}^{2} + 25$$ T3^4 + 11*T3^2 + 25 $$T_{7}^{2} + 4$$ T7^2 + 4 $$T_{11}^{2} - 3T_{11} - 3$$ T11^2 - 3*T11 - 3 $$T_{13}^{2} + 3T_{13} - 3$$ T13^2 + 3*T13 - 3

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$(T - 1)^{4}$$
$3$ $$T^{4} + 11T^{2} + 25$$
$5$ $$T^{4}$$
$7$ $$(T^{2} + 4)^{2}$$
$11$ $$(T^{2} - 3 T - 3)^{2}$$
$13$ $$(T^{2} + 3 T - 3)^{2}$$
$17$ $$(T^{2} + 6 T - 12)^{2}$$
$19$ $$T^{4} + 60T^{2} + 144$$
$23$ $$(T^{2} + 3 T - 3)^{2}$$
$29$ $$T^{4} + 15T^{2} + 9$$
$31$ $$T^{4} + 99T^{2} + 2025$$
$37$ $$T^{4} - 10T^{2} + 1369$$
$41$ $$(T^{2} - 15 T + 51)^{2}$$
$43$ $$(T + 6)^{4}$$
$47$ $$T^{4} + 60T^{2} + 144$$
$53$ $$T^{4} + 60T^{2} + 144$$
$59$ $$T^{4} + 60T^{2} + 144$$
$61$ $$T^{4} + 231 T^{2} + 11025$$
$67$ $$T^{4} + 95T^{2} + 2209$$
$71$ $$(T^{2} - 84)^{2}$$
$73$ $$T^{4} + 107T^{2} + 1681$$
$79$ $$T^{4} + 231 T^{2} + 11025$$
$83$ $$T^{4} + 240T^{2} + 2304$$
$89$ $$(T^{2} + 36)^{2}$$
$97$ $$(T^{2} - 18 T + 60)^{2}$$
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