# Properties

 Label 1232.4.a.f 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

 $$f(q)$$ $$=$$ $$q + 2 q^{3} + 18 q^{5} - 7 q^{7} - 23 q^{9}+O(q^{10})$$ q + 2 * q^3 + 18 * q^5 - 7 * q^7 - 23 * q^9 $$q + 2 q^{3} + 18 q^{5} - 7 q^{7} - 23 q^{9} + 11 q^{11} + 56 q^{13} + 36 q^{15} + 36 q^{17} + 28 q^{19} - 14 q^{21} - 180 q^{23} + 199 q^{25} - 100 q^{27} - 54 q^{29} + 334 q^{31} + 22 q^{33} - 126 q^{35} + 386 q^{37} + 112 q^{39} - 444 q^{41} + 316 q^{43} - 414 q^{45} + 402 q^{47} + 49 q^{49} + 72 q^{51} - 486 q^{53} + 198 q^{55} + 56 q^{57} + 282 q^{59} + 380 q^{61} + 161 q^{63} + 1008 q^{65} - 176 q^{67} - 360 q^{69} + 324 q^{71} + 800 q^{73} + 398 q^{75} - 77 q^{77} + 1144 q^{79} + 421 q^{81} - 468 q^{83} + 648 q^{85} - 108 q^{87} - 870 q^{89} - 392 q^{91} + 668 q^{93} + 504 q^{95} - 1330 q^{97} - 253 q^{99}+O(q^{100})$$ q + 2 * q^3 + 18 * q^5 - 7 * q^7 - 23 * q^9 + 11 * q^11 + 56 * q^13 + 36 * q^15 + 36 * q^17 + 28 * q^19 - 14 * q^21 - 180 * q^23 + 199 * q^25 - 100 * q^27 - 54 * q^29 + 334 * q^31 + 22 * q^33 - 126 * q^35 + 386 * q^37 + 112 * q^39 - 444 * q^41 + 316 * q^43 - 414 * q^45 + 402 * q^47 + 49 * q^49 + 72 * q^51 - 486 * q^53 + 198 * q^55 + 56 * q^57 + 282 * q^59 + 380 * q^61 + 161 * q^63 + 1008 * q^65 - 176 * q^67 - 360 * q^69 + 324 * q^71 + 800 * q^73 + 398 * q^75 - 77 * q^77 + 1144 * q^79 + 421 * q^81 - 468 * q^83 + 648 * q^85 - 108 * q^87 - 870 * q^89 - 392 * q^91 + 668 * q^93 + 504 * q^95 - 1330 * q^97 - 253 * 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 2.00000 0 18.0000 0 −7.00000 0 −23.0000 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.f 1
4.b odd 2 1 154.4.a.d 1
12.b even 2 1 1386.4.a.a 1
28.d even 2 1 1078.4.a.g 1
44.c even 2 1 1694.4.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
154.4.a.d 1 4.b odd 2 1
1078.4.a.g 1 28.d even 2 1
1232.4.a.f 1 1.a even 1 1 trivial
1386.4.a.a 1 12.b even 2 1
1694.4.a.c 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 $$S_{4}^{\mathrm{new}}(\Gamma_0(1232))$$:

 $$T_{3} - 2$$ T3 - 2 $$T_{5} - 18$$ T5 - 18

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$T - 2$$
$5$ $$T - 18$$
$7$ $$T + 7$$
$11$ $$T - 11$$
$13$ $$T - 56$$
$17$ $$T - 36$$
$19$ $$T - 28$$
$23$ $$T + 180$$
$29$ $$T + 54$$
$31$ $$T - 334$$
$37$ $$T - 386$$
$41$ $$T + 444$$
$43$ $$T - 316$$
$47$ $$T - 402$$
$53$ $$T + 486$$
$59$ $$T - 282$$
$61$ $$T - 380$$
$67$ $$T + 176$$
$71$ $$T - 324$$
$73$ $$T - 800$$
$79$ $$T - 1144$$
$83$ $$T + 468$$
$89$ $$T + 870$$
$97$ $$T + 1330$$