# Properties

 Label 3822.2.c.e Level $3822$ Weight $2$ Character orbit 3822.c Analytic conductor $30.519$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3822 = 2 \cdot 3 \cdot 7^{2} \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3822.c (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.5188236525$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-1})$$ Defining polynomial: $$x^{2} + 1$$ x^2 + 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 546) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$i = \sqrt{-1}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + i q^{2} + q^{3} - q^{4} + i q^{5} + i q^{6} - i q^{8} + q^{9} +O(q^{10})$$ q + i * q^2 + q^3 - q^4 + i * q^5 + i * q^6 - i * q^8 + q^9 $$q + i q^{2} + q^{3} - q^{4} + i q^{5} + i q^{6} - i q^{8} + q^{9} - q^{10} - i q^{11} - q^{12} + (2 i - 3) q^{13} + i q^{15} + q^{16} - q^{17} + i q^{18} - i q^{19} - i q^{20} + q^{22} + 3 q^{23} - i q^{24} + 4 q^{25} + ( - 3 i - 2) q^{26} + q^{27} + 9 q^{29} - q^{30} - 4 i q^{31} + i q^{32} - i q^{33} - i q^{34} - q^{36} + 9 i q^{37} + q^{38} + (2 i - 3) q^{39} + q^{40} + 8 i q^{41} + 7 q^{43} + i q^{44} + i q^{45} + 3 i q^{46} + 8 i q^{47} + q^{48} + 4 i q^{50} - q^{51} + ( - 2 i + 3) q^{52} - 10 q^{53} + i q^{54} + q^{55} - i q^{57} + 9 i q^{58} - 6 i q^{59} - i q^{60} - 11 q^{61} + 4 q^{62} - q^{64} + ( - 3 i - 2) q^{65} + q^{66} + 12 i q^{67} + q^{68} + 3 q^{69} - 6 i q^{71} - i q^{72} + 11 i q^{73} - 9 q^{74} + 4 q^{75} + i q^{76} + ( - 3 i - 2) q^{78} - 12 q^{79} + i q^{80} + q^{81} - 8 q^{82} + 6 i q^{83} - i q^{85} + 7 i q^{86} + 9 q^{87} - q^{88} + 12 i q^{89} - q^{90} - 3 q^{92} - 4 i q^{93} - 8 q^{94} + q^{95} + i q^{96} - 2 i q^{97} - i q^{99} +O(q^{100})$$ q + i * q^2 + q^3 - q^4 + i * q^5 + i * q^6 - i * q^8 + q^9 - q^10 - i * q^11 - q^12 + (2*i - 3) * q^13 + i * q^15 + q^16 - q^17 + i * q^18 - i * q^19 - i * q^20 + q^22 + 3 * q^23 - i * q^24 + 4 * q^25 + (-3*i - 2) * q^26 + q^27 + 9 * q^29 - q^30 - 4*i * q^31 + i * q^32 - i * q^33 - i * q^34 - q^36 + 9*i * q^37 + q^38 + (2*i - 3) * q^39 + q^40 + 8*i * q^41 + 7 * q^43 + i * q^44 + i * q^45 + 3*i * q^46 + 8*i * q^47 + q^48 + 4*i * q^50 - q^51 + (-2*i + 3) * q^52 - 10 * q^53 + i * q^54 + q^55 - i * q^57 + 9*i * q^58 - 6*i * q^59 - i * q^60 - 11 * q^61 + 4 * q^62 - q^64 + (-3*i - 2) * q^65 + q^66 + 12*i * q^67 + q^68 + 3 * q^69 - 6*i * q^71 - i * q^72 + 11*i * q^73 - 9 * q^74 + 4 * q^75 + i * q^76 + (-3*i - 2) * q^78 - 12 * q^79 + i * q^80 + q^81 - 8 * q^82 + 6*i * q^83 - i * q^85 + 7*i * q^86 + 9 * q^87 - q^88 + 12*i * q^89 - q^90 - 3 * q^92 - 4*i * q^93 - 8 * q^94 + q^95 + i * q^96 - 2*i * q^97 - i * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9 $$2 q + 2 q^{3} - 2 q^{4} + 2 q^{9} - 2 q^{10} - 2 q^{12} - 6 q^{13} + 2 q^{16} - 2 q^{17} + 2 q^{22} + 6 q^{23} + 8 q^{25} - 4 q^{26} + 2 q^{27} + 18 q^{29} - 2 q^{30} - 2 q^{36} + 2 q^{38} - 6 q^{39} + 2 q^{40} + 14 q^{43} + 2 q^{48} - 2 q^{51} + 6 q^{52} - 20 q^{53} + 2 q^{55} - 22 q^{61} + 8 q^{62} - 2 q^{64} - 4 q^{65} + 2 q^{66} + 2 q^{68} + 6 q^{69} - 18 q^{74} + 8 q^{75} - 4 q^{78} - 24 q^{79} + 2 q^{81} - 16 q^{82} + 18 q^{87} - 2 q^{88} - 2 q^{90} - 6 q^{92} - 16 q^{94} + 2 q^{95}+O(q^{100})$$ 2 * q + 2 * q^3 - 2 * q^4 + 2 * q^9 - 2 * q^10 - 2 * q^12 - 6 * q^13 + 2 * q^16 - 2 * q^17 + 2 * q^22 + 6 * q^23 + 8 * q^25 - 4 * q^26 + 2 * q^27 + 18 * q^29 - 2 * q^30 - 2 * q^36 + 2 * q^38 - 6 * q^39 + 2 * q^40 + 14 * q^43 + 2 * q^48 - 2 * q^51 + 6 * q^52 - 20 * q^53 + 2 * q^55 - 22 * q^61 + 8 * q^62 - 2 * q^64 - 4 * q^65 + 2 * q^66 + 2 * q^68 + 6 * q^69 - 18 * q^74 + 8 * q^75 - 4 * q^78 - 24 * q^79 + 2 * q^81 - 16 * q^82 + 18 * q^87 - 2 * q^88 - 2 * q^90 - 6 * q^92 - 16 * q^94 + 2 * q^95

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3822\mathbb{Z}\right)^\times$$.

 $$n$$ $$1471$$ $$2549$$ $$3433$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
883.1
 − 1.00000i 1.00000i
1.00000i 1.00000 −1.00000 1.00000i 1.00000i 0 1.00000i 1.00000 −1.00000
883.2 1.00000i 1.00000 −1.00000 1.00000i 1.00000i 0 1.00000i 1.00000 −1.00000
 $$n$$: 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3822.2.c.e 2
7.b odd 2 1 546.2.c.a 2
13.b even 2 1 inner 3822.2.c.e 2
21.c even 2 1 1638.2.c.f 2
28.d even 2 1 4368.2.h.k 2
91.b odd 2 1 546.2.c.a 2
91.i even 4 1 7098.2.a.d 1
91.i even 4 1 7098.2.a.s 1
273.g even 2 1 1638.2.c.f 2
364.h even 2 1 4368.2.h.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.c.a 2 7.b odd 2 1
546.2.c.a 2 91.b odd 2 1
1638.2.c.f 2 21.c even 2 1
1638.2.c.f 2 273.g even 2 1
3822.2.c.e 2 1.a even 1 1 trivial
3822.2.c.e 2 13.b even 2 1 inner
4368.2.h.k 2 28.d even 2 1
4368.2.h.k 2 364.h even 2 1
7098.2.a.d 1 91.i even 4 1
7098.2.a.s 1 91.i even 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3822, [\chi])$$:

 $$T_{5}^{2} + 1$$ T5^2 + 1 $$T_{11}^{2} + 1$$ T11^2 + 1 $$T_{17} + 1$$ T17 + 1 $$T_{19}^{2} + 1$$ T19^2 + 1

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$(T - 1)^{2}$$
$5$ $$T^{2} + 1$$
$7$ $$T^{2}$$
$11$ $$T^{2} + 1$$
$13$ $$T^{2} + 6T + 13$$
$17$ $$(T + 1)^{2}$$
$19$ $$T^{2} + 1$$
$23$ $$(T - 3)^{2}$$
$29$ $$(T - 9)^{2}$$
$31$ $$T^{2} + 16$$
$37$ $$T^{2} + 81$$
$41$ $$T^{2} + 64$$
$43$ $$(T - 7)^{2}$$
$47$ $$T^{2} + 64$$
$53$ $$(T + 10)^{2}$$
$59$ $$T^{2} + 36$$
$61$ $$(T + 11)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$T^{2} + 36$$
$73$ $$T^{2} + 121$$
$79$ $$(T + 12)^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$T^{2} + 144$$
$97$ $$T^{2} + 4$$