# Properties

 Label 4312.2.a.h Level $4312$ Weight $2$ Character orbit 4312.a Self dual yes Analytic conductor $34.431$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4312 = 2^{3} \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4312.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$34.4314933516$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 616) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{3} + q^{5} - 2 q^{9}+O(q^{10})$$ q + q^3 + q^5 - 2 * q^9 $$q + q^{3} + q^{5} - 2 q^{9} + q^{11} + q^{15} + 2 q^{17} + 2 q^{19} - 7 q^{23} - 4 q^{25} - 5 q^{27} - 10 q^{29} - 7 q^{31} + q^{33} - 9 q^{37} + 2 q^{41} - 4 q^{43} - 2 q^{45} - 8 q^{47} + 2 q^{51} + 2 q^{53} + q^{55} + 2 q^{57} + 15 q^{59} + 14 q^{61} + 3 q^{67} - 7 q^{69} + 3 q^{71} - 10 q^{73} - 4 q^{75} + 10 q^{79} + q^{81} + 2 q^{85} - 10 q^{87} + 11 q^{89} - 7 q^{93} + 2 q^{95} - 7 q^{97} - 2 q^{99}+O(q^{100})$$ q + q^3 + q^5 - 2 * q^9 + q^11 + q^15 + 2 * q^17 + 2 * q^19 - 7 * q^23 - 4 * q^25 - 5 * q^27 - 10 * q^29 - 7 * q^31 + q^33 - 9 * q^37 + 2 * q^41 - 4 * q^43 - 2 * q^45 - 8 * q^47 + 2 * q^51 + 2 * q^53 + q^55 + 2 * q^57 + 15 * q^59 + 14 * q^61 + 3 * q^67 - 7 * q^69 + 3 * q^71 - 10 * q^73 - 4 * q^75 + 10 * q^79 + q^81 + 2 * q^85 - 10 * q^87 + 11 * q^89 - 7 * q^93 + 2 * q^95 - 7 * q^97 - 2 * 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
0 1.00000 0 1.00000 0 0 0 −2.00000 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
$$2$$ $$1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 4312.2.a.h 1
4.b odd 2 1 8624.2.a.m 1
7.b odd 2 1 616.2.a.b 1
21.c even 2 1 5544.2.a.o 1
28.d even 2 1 1232.2.a.i 1
56.e even 2 1 4928.2.a.k 1
56.h odd 2 1 4928.2.a.ba 1
77.b even 2 1 6776.2.a.d 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
616.2.a.b 1 7.b odd 2 1
1232.2.a.i 1 28.d even 2 1
4312.2.a.h 1 1.a even 1 1 trivial
4928.2.a.k 1 56.e even 2 1
4928.2.a.ba 1 56.h odd 2 1
5544.2.a.o 1 21.c even 2 1
6776.2.a.d 1 77.b even 2 1
8624.2.a.m 1 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4312))$$:

 $$T_{3} - 1$$ T3 - 1 $$T_{5} - 1$$ T5 - 1 $$T_{13}$$ T13 $$T_{17} - 2$$ T17 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 1$$
$5$ $$T - 1$$
$7$ $$T$$
$11$ $$T - 1$$
$13$ $$T$$
$17$ $$T - 2$$
$19$ $$T - 2$$
$23$ $$T + 7$$
$29$ $$T + 10$$
$31$ $$T + 7$$
$37$ $$T + 9$$
$41$ $$T - 2$$
$43$ $$T + 4$$
$47$ $$T + 8$$
$53$ $$T - 2$$
$59$ $$T - 15$$
$61$ $$T - 14$$
$67$ $$T - 3$$
$71$ $$T - 3$$
$73$ $$T + 10$$
$79$ $$T - 10$$
$83$ $$T$$
$89$ $$T - 11$$
$97$ $$T + 7$$