Properties

 Label 3850.2.a.f Level $3850$ Weight $2$ Character orbit 3850.a Self dual yes Analytic conductor $30.742$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$3850 = 2 \cdot 5^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3850.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$30.7424047782$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 154) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
−1.00000 0 1.00000 0 0 1.00000 −1.00000 −3.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$1$$
$$7$$ $$-1$$
$$11$$ $$1$$

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3850.2.a.f 1
5.b even 2 1 154.2.a.c 1
5.c odd 4 2 3850.2.c.l 2
15.d odd 2 1 1386.2.a.b 1
20.d odd 2 1 1232.2.a.h 1
35.c odd 2 1 1078.2.a.j 1
35.i odd 6 2 1078.2.e.c 2
35.j even 6 2 1078.2.e.b 2
40.e odd 2 1 4928.2.a.o 1
40.f even 2 1 4928.2.a.n 1
55.d odd 2 1 1694.2.a.c 1
105.g even 2 1 9702.2.a.v 1
140.c even 2 1 8624.2.a.o 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.2.a.c 1 5.b even 2 1
1078.2.a.j 1 35.c odd 2 1
1078.2.e.b 2 35.j even 6 2
1078.2.e.c 2 35.i odd 6 2
1232.2.a.h 1 20.d odd 2 1
1386.2.a.b 1 15.d odd 2 1
1694.2.a.c 1 55.d odd 2 1
3850.2.a.f 1 1.a even 1 1 trivial
3850.2.c.l 2 5.c odd 4 2
4928.2.a.n 1 40.f even 2 1
4928.2.a.o 1 40.e odd 2 1
8624.2.a.o 1 140.c even 2 1
9702.2.a.v 1 105.g even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(3850))$$:

 $$T_{3}$$ T3 $$T_{13} + 2$$ T13 + 2 $$T_{17} + 2$$ T17 + 2 $$T_{19}$$ T19

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T + 1$$
$13$ $$T + 2$$
$17$ $$T + 2$$
$19$ $$T$$
$23$ $$T - 8$$
$29$ $$T + 2$$
$31$ $$T + 8$$
$37$ $$T - 2$$
$41$ $$T - 10$$
$43$ $$T + 4$$
$47$ $$T + 8$$
$53$ $$T + 6$$
$59$ $$T$$
$61$ $$T - 10$$
$67$ $$T - 12$$
$71$ $$T - 16$$
$73$ $$T - 14$$
$79$ $$T$$
$83$ $$T$$
$89$ $$T + 6$$
$97$ $$T + 10$$