# Properties

 Label 1232.4.a.i Level $1232$ Weight $4$ Character orbit 1232.a Self dual yes Analytic conductor $72.690$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1232 = 2^{4} \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1232.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$72.6903531271$$ 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

 $$f(q)$$ $$=$$ $$q + 10 q^{3} - 14 q^{5} - 7 q^{7} + 73 q^{9}+O(q^{10})$$ q + 10 * q^3 - 14 * q^5 - 7 * q^7 + 73 * q^9 $$q + 10 q^{3} - 14 q^{5} - 7 q^{7} + 73 q^{9} + 11 q^{11} - 16 q^{13} - 140 q^{15} + 108 q^{17} - 116 q^{19} - 70 q^{21} - 68 q^{23} + 71 q^{25} + 460 q^{27} + 122 q^{29} + 262 q^{31} + 110 q^{33} + 98 q^{35} + 130 q^{37} - 160 q^{39} + 204 q^{41} + 396 q^{43} - 1022 q^{45} - 166 q^{47} + 49 q^{49} + 1080 q^{51} + 442 q^{53} - 154 q^{55} - 1160 q^{57} - 702 q^{59} + 196 q^{61} - 511 q^{63} + 224 q^{65} + 416 q^{67} - 680 q^{69} - 492 q^{71} + 408 q^{73} + 710 q^{75} - 77 q^{77} - 600 q^{79} + 2629 q^{81} + 1212 q^{83} - 1512 q^{85} + 1220 q^{87} + 1146 q^{89} + 112 q^{91} + 2620 q^{93} + 1624 q^{95} - 482 q^{97} + 803 q^{99}+O(q^{100})$$ q + 10 * q^3 - 14 * q^5 - 7 * q^7 + 73 * q^9 + 11 * q^11 - 16 * q^13 - 140 * q^15 + 108 * q^17 - 116 * q^19 - 70 * q^21 - 68 * q^23 + 71 * q^25 + 460 * q^27 + 122 * q^29 + 262 * q^31 + 110 * q^33 + 98 * q^35 + 130 * q^37 - 160 * q^39 + 204 * q^41 + 396 * q^43 - 1022 * q^45 - 166 * q^47 + 49 * q^49 + 1080 * q^51 + 442 * q^53 - 154 * q^55 - 1160 * q^57 - 702 * q^59 + 196 * q^61 - 511 * q^63 + 224 * q^65 + 416 * q^67 - 680 * q^69 - 492 * q^71 + 408 * q^73 + 710 * q^75 - 77 * q^77 - 600 * q^79 + 2629 * q^81 + 1212 * q^83 - 1512 * q^85 + 1220 * q^87 + 1146 * q^89 + 112 * q^91 + 2620 * q^93 + 1624 * q^95 - 482 * q^97 + 803 * 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 10.0000 0 −14.0000 0 −7.00000 0 73.0000 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 1232.4.a.i 1
4.b odd 2 1 154.4.a.c 1
12.b even 2 1 1386.4.a.g 1
28.d even 2 1 1078.4.a.h 1
44.c even 2 1 1694.4.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.4.a.c 1 4.b odd 2 1
1078.4.a.h 1 28.d even 2 1
1232.4.a.i 1 1.a even 1 1 trivial
1386.4.a.g 1 12.b even 2 1
1694.4.a.a 1 44.c even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{4}^{\mathrm{new}}(\Gamma_0(1232))$$:

 $$T_{3} - 10$$ T3 - 10 $$T_{5} + 14$$ T5 + 14

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 10$$
$5$ $$T + 14$$
$7$ $$T + 7$$
$11$ $$T - 11$$
$13$ $$T + 16$$
$17$ $$T - 108$$
$19$ $$T + 116$$
$23$ $$T + 68$$
$29$ $$T - 122$$
$31$ $$T - 262$$
$37$ $$T - 130$$
$41$ $$T - 204$$
$43$ $$T - 396$$
$47$ $$T + 166$$
$53$ $$T - 442$$
$59$ $$T + 702$$
$61$ $$T - 196$$
$67$ $$T - 416$$
$71$ $$T + 492$$
$73$ $$T - 408$$
$79$ $$T + 600$$
$83$ $$T - 1212$$
$89$ $$T - 1146$$
$97$ $$T + 482$$