# Properties

 Label 3850.2.c.j Level $3850$ Weight $2$ Character orbit 3850.c Analytic conductor $30.742$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3850.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.7424047782$$ 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 154) 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.

 $$f(q)$$ $$=$$ $$q - i q^{2} - q^{4} - i q^{7} + i q^{8} + 3 q^{9} +O(q^{10})$$ q - i * q^2 - q^4 - i * q^7 + i * q^8 + 3 * q^9 $$q - i q^{2} - q^{4} - i q^{7} + i q^{8} + 3 q^{9} - q^{11} - 2 i q^{13} - q^{14} + q^{16} - 4 i q^{17} - 3 i q^{18} + 6 q^{19} + i q^{22} - 4 i q^{23} - 2 q^{26} + i q^{28} + 2 q^{29} - 2 q^{31} - i q^{32} - 4 q^{34} - 3 q^{36} + 10 i q^{37} - 6 i q^{38} + 4 q^{41} + 8 i q^{43} + q^{44} - 4 q^{46} + 2 i q^{47} - q^{49} + 2 i q^{52} - 6 i q^{53} + q^{56} - 2 i q^{58} + 12 q^{59} - 14 q^{61} + 2 i q^{62} - 3 i q^{63} - q^{64} - 12 i q^{67} + 4 i q^{68} - 8 q^{71} + 3 i q^{72} - 4 i q^{73} + 10 q^{74} - 6 q^{76} + i q^{77} + 9 q^{81} - 4 i q^{82} + 6 i q^{83} + 8 q^{86} - i q^{88} + 6 q^{89} - 2 q^{91} + 4 i q^{92} + 2 q^{94} - 14 i q^{97} + i q^{98} - 3 q^{99} +O(q^{100})$$ q - i * q^2 - q^4 - i * q^7 + i * q^8 + 3 * q^9 - q^11 - 2*i * q^13 - q^14 + q^16 - 4*i * q^17 - 3*i * q^18 + 6 * q^19 + i * q^22 - 4*i * q^23 - 2 * q^26 + i * q^28 + 2 * q^29 - 2 * q^31 - i * q^32 - 4 * q^34 - 3 * q^36 + 10*i * q^37 - 6*i * q^38 + 4 * q^41 + 8*i * q^43 + q^44 - 4 * q^46 + 2*i * q^47 - q^49 + 2*i * q^52 - 6*i * q^53 + q^56 - 2*i * q^58 + 12 * q^59 - 14 * q^61 + 2*i * q^62 - 3*i * q^63 - q^64 - 12*i * q^67 + 4*i * q^68 - 8 * q^71 + 3*i * q^72 - 4*i * q^73 + 10 * q^74 - 6 * q^76 + i * q^77 + 9 * q^81 - 4*i * q^82 + 6*i * q^83 + 8 * q^86 - i * q^88 + 6 * q^89 - 2 * q^91 + 4*i * q^92 + 2 * q^94 - 14*i * q^97 + i * q^98 - 3 * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{4} + 6 q^{9}+O(q^{10})$$ 2 * q - 2 * q^4 + 6 * q^9 $$2 q - 2 q^{4} + 6 q^{9} - 2 q^{11} - 2 q^{14} + 2 q^{16} + 12 q^{19} - 4 q^{26} + 4 q^{29} - 4 q^{31} - 8 q^{34} - 6 q^{36} + 8 q^{41} + 2 q^{44} - 8 q^{46} - 2 q^{49} + 2 q^{56} + 24 q^{59} - 28 q^{61} - 2 q^{64} - 16 q^{71} + 20 q^{74} - 12 q^{76} + 18 q^{81} + 16 q^{86} + 12 q^{89} - 4 q^{91} + 4 q^{94} - 6 q^{99}+O(q^{100})$$ 2 * q - 2 * q^4 + 6 * q^9 - 2 * q^11 - 2 * q^14 + 2 * q^16 + 12 * q^19 - 4 * q^26 + 4 * q^29 - 4 * q^31 - 8 * q^34 - 6 * q^36 + 8 * q^41 + 2 * q^44 - 8 * q^46 - 2 * q^49 + 2 * q^56 + 24 * q^59 - 28 * q^61 - 2 * q^64 - 16 * q^71 + 20 * q^74 - 12 * q^76 + 18 * q^81 + 16 * q^86 + 12 * q^89 - 4 * q^91 + 4 * q^94 - 6 * q^99

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/3850\mathbb{Z}\right)^\times$$.

 $$n$$ $$1751$$ $$2201$$ $$2927$$ $$\chi(n)$$ $$1$$ $$1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1849.1
 1.00000i − 1.00000i
1.00000i 0 −1.00000 0 0 1.00000i 1.00000i 3.00000 0
1849.2 1.00000i 0 −1.00000 0 0 1.00000i 1.00000i 3.00000 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 3850.2.c.j 2
5.b even 2 1 inner 3850.2.c.j 2
5.c odd 4 1 154.2.a.a 1
5.c odd 4 1 3850.2.a.u 1
15.e even 4 1 1386.2.a.l 1
20.e even 4 1 1232.2.a.e 1
35.f even 4 1 1078.2.a.d 1
35.k even 12 2 1078.2.e.i 2
35.l odd 12 2 1078.2.e.j 2
40.i odd 4 1 4928.2.a.v 1
40.k even 4 1 4928.2.a.w 1
55.e even 4 1 1694.2.a.g 1
105.k odd 4 1 9702.2.a.ba 1
140.j odd 4 1 8624.2.a.r 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.a 1 5.c odd 4 1
1078.2.a.d 1 35.f even 4 1
1078.2.e.i 2 35.k even 12 2
1078.2.e.j 2 35.l odd 12 2
1232.2.a.e 1 20.e even 4 1
1386.2.a.l 1 15.e even 4 1
1694.2.a.g 1 55.e even 4 1
3850.2.a.u 1 5.c odd 4 1
3850.2.c.j 2 1.a even 1 1 trivial
3850.2.c.j 2 5.b even 2 1 inner
4928.2.a.v 1 40.i odd 4 1
4928.2.a.w 1 40.k even 4 1
8624.2.a.r 1 140.j odd 4 1
9702.2.a.ba 1 105.k odd 4 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3850, [\chi])$$:

 $$T_{3}$$ T3 $$T_{13}^{2} + 4$$ T13^2 + 4 $$T_{17}^{2} + 16$$ T17^2 + 16 $$T_{19} - 6$$ T19 - 6 $$T_{37}^{2} + 100$$ T37^2 + 100

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2} + 1$$
$3$ $$T^{2}$$
$5$ $$T^{2}$$
$7$ $$T^{2} + 1$$
$11$ $$(T + 1)^{2}$$
$13$ $$T^{2} + 4$$
$17$ $$T^{2} + 16$$
$19$ $$(T - 6)^{2}$$
$23$ $$T^{2} + 16$$
$29$ $$(T - 2)^{2}$$
$31$ $$(T + 2)^{2}$$
$37$ $$T^{2} + 100$$
$41$ $$(T - 4)^{2}$$
$43$ $$T^{2} + 64$$
$47$ $$T^{2} + 4$$
$53$ $$T^{2} + 36$$
$59$ $$(T - 12)^{2}$$
$61$ $$(T + 14)^{2}$$
$67$ $$T^{2} + 144$$
$71$ $$(T + 8)^{2}$$
$73$ $$T^{2} + 16$$
$79$ $$T^{2}$$
$83$ $$T^{2} + 36$$
$89$ $$(T - 6)^{2}$$
$97$ $$T^{2} + 196$$